Can Optimized Portfolios Beat 1 - University of Nottingham · Can Optimized Portfolios Beat 1/N?...

Can Optimized Portfolios Beat 1/N?

This dissertation is presented in part fulfillmentof the requirement for the completion of an

MSc in Economicsin the Department of Economics, University of Konstanz,

and anMSc in Economics and Econometrics

in the School of Economics, University of Nottingham.The work is the sole responsibility of the candidate.

By:

Period of Completion:

Word Count:

1st Assessor:2nd Assessor:

Valerius Disch

April 21, 2018 - August 21, 2018

14,777 / 15,000

Professor Marcel Fischer, University of Konstanz Professor Patrick Marsh, University of Nottingham

Konstanz, August 21, 2018

Abstract ii

Abstract

This analysis conducts an out-of-sample horse race between the naive diversification ap-proach and 15 optimized portfolio strategies across four US and two European equity datasets. Not a single optimized strategy achieves to consistently outperform the naive diver-sification benchmark in a statistically significant manner based on a total of six perfor-mance evaluation criteria. However, unconstrained strategies related to the sample-basedmean-variance and minimum-variance strategy exhibit superior performances in two USdata sets. Timing strategies attain consistently good results for US industry data sets.Performance behavior appears to be sensitive to respective data sets, yet independent ofperformance evaluation criteria.

Contents iii

ContentsAbstract ii

List of Figures v

List of Tables v

List of Abbreviations vi

1. Introduction 1

2. Literature Review 2

3. Data 4

4. Optimized Portfolio Strategies and 1/N 104.1. Mean-Variance Framework . . . . . . . . . . . . . . . . . . . . . . . . . 104.2. Estimation Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.3. Approaches to Fight Estimation Risk . . . . . . . . . . . . . . . . . . . . 144.4. Portfolio Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.4.1. Equally-Weighted . . . . . . . . . . . . . . . . . . . . . . . . . 174.4.2. Sample-Based Mean-Variance . . . . . . . . . . . . . . . . . . . 204.4.3. Minimum-Variance . . . . . . . . . . . . . . . . . . . . . . . . . 204.4.4. Value-Weighted . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.4.5. MacKinlay-Pástor . . . . . . . . . . . . . . . . . . . . . . . . . 214.4.6. Bayes-Stein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.4.7. Portfolio Weight Constraints . . . . . . . . . . . . . . . . . . . . 234.4.8. Mean-Variance and Minimum-Variance . . . . . . . . . . . . . . 244.4.9. Equally-Weighted and Minimum-Variance . . . . . . . . . . . . 254.4.10. Equally-Weighted and Mean-Variance . . . . . . . . . . . . . . . 264.4.11. Equally-Weighted and Mean-Variance and Minimum-Variance . . 274.4.12. Volatility Timing . . . . . . . . . . . . . . . . . . . . . . . . . . 284.4.13. Reward-to-Risk Timing . . . . . . . . . . . . . . . . . . . . . . 28

5. Methodology 295.1. Estimation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.2. Performance Evaluation Criteria . . . . . . . . . . . . . . . . . . . . . . 30

5.2.1. Sharpe Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.2.2. Sortino Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.2.3. Omega Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.2.4. Calmar Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.2.5. Return on Value-at-Risk . . . . . . . . . . . . . . . . . . . . . . 345.2.6. Certainty Equivalent Return . . . . . . . . . . . . . . . . . . . . 35

6. Empirical Results and Discussion 36

7. Robustness Checks 467.1. Estimation Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467.2. Risk-Aversion Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . 477.3. Tuning Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Contents iv

8. Conclusion 48

Appendices 49

Appendix A. Details of the Data Sets 49A.1. US MKT, PF6, PF25, IND10, and IND48 . . . . . . . . . . . . . . . . . . 49A.2. European MKT, PF6, and PF25 . . . . . . . . . . . . . . . . . . . . . . . 50

Appendix B. Mean-Variance Optimization in Excess Returns 51

Appendix C. Minimum-Variance Optimization 52

Appendix D. Robustness Checks Tables 53

References 64

List of Figures v

List of Figures3.1. Boxplots for the US and European MKT Series and PF6 Portfolios . . . . 73.2. Monthly Cumulative MKT Series for the US and Europe . . . . . . . . . 83.3. Histogram, Normal Density, and Kernel Density Estimate for the US and

European MKT Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.4. Quantile-Quantile Plot for the US and European MKT Series . . . . . . . 94.1. Effects of Naive Diversification on the Portfolio’s Standard Deviation . . 19

List of Tables3.1. Overview Data Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2. Descriptive Statistics for the US and European MKT Series and PF6 Port-

folios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.3. Correlation Table for the US and European MKT Series and PF6 Portfolios 73.4. Normality and Stationarity Tests for the US and European MKT Series . . 94.1. Overview Portfolio Strategies . . . . . . . . . . . . . . . . . . . . . . . . 166.1. Sharpe Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.2. Average Minimum and Maximum Sharpe Ratio Portfolio Weights . . . . 396.3. Sortino and Omega Ratios . . . . . . . . . . . . . . . . . . . . . . . . . 416.4. Calmar Ratios and Returns on Value-at-Risk . . . . . . . . . . . . . . . . 436.5. Certainty Equivalent Returns . . . . . . . . . . . . . . . . . . . . . . . . 446.6. Performance Evaluation Criteria Rank Correlations . . . . . . . . . . . . 456.7. Data Sets Rank Correlations . . . . . . . . . . . . . . . . . . . . . . . . 45D.1. Sharpe Ratios, Estimation Window M = 60 . . . . . . . . . . . . . . . . 53D.2. Certainty Equivalent Returns, Estimation Window M = 60 . . . . . . . . 54D.3. Sharpe Ratios, Estimation Window M = 180 . . . . . . . . . . . . . . . . 55D.4. Certainty Equivalent Returns, Estimation Window M = 180 . . . . . . . . 56D.5. Sharpe Ratios, Expanding Window . . . . . . . . . . . . . . . . . . . . . 57D.6. Certainty Equivalent Returns, Expanding Window . . . . . . . . . . . . . 58D.7. Sharpe Ratios, Risk-Aversion Parameter γ = 2 . . . . . . . . . . . . . . . 59D.8. Certainty Equivalent Returns, Risk-Aversion Parameter γ = 2 . . . . . . . 60D.9. Sharpe Ratios, Risk-Aversion Parameter γ = 4 . . . . . . . . . . . . . . . 61D.10.Certainty Equivalent Returns, Risk-Aversion Parameter γ = 4 . . . . . . . 62D.11.Sharpe Ratios and Certainty Equivalent Returns, Tuning Parameter η = 2

and η = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

List of Abbreviations vi

List of AbbreviationsAMEX = American Stock ExchangeAR(p) = Autoregressive Model of Order pBS = Bayes-SteinBSsc = Bayes-Stein Short Sale ConstraintCAPM = Capital Asset Pricing ModelCE = Certainty Equivalent ReturnCR = Calmar RatioDD = Drawdownet al. = et alia, “and others”EU = EuropeEW = Equally-WeightedEW/MinV = Equally-Weighted and Minimum-VarianceEW/MV = Equally-Weighted and Mean-VarianceEW/MV/MinV = Equally-Weighted and Mean-Variance and Minimum-VarianceHAC = Heteroskedasticity and Autocorrelationi.e. = id est, “that is to say”IS = In-SampleJK = Jobson and KorkieLPM = Lower Partial MomentsLW = Ledoit and WolfMAR = Minimum Acceptable ReturnMaxDD = Maximum DrawdownMinV = Minimum-VarianceMinVsc = Minimum-Variance Short Sale ConstraintMKT = Excess Return on the MarketML = Maximum LikelihoodMP = MacKinlay-PástorMSCI = Morgan Stanley Capital InternationalMV = Mean-VarianceMVlsc = Mean-Variance Long-Short Sale ConstraintMV/MinV = Mean-Variance and Minimum-VarianceMVsc = Mean-Variance Short Sale ConstraintNASDAQ = National Association of Securities Dealers Automated QuotationsNYSE = New York Stock ExchangeOR = Omega RatioRoVaR = Return on Value-at-RiskRRT = Reward-to-Risk TimingSoR = Sortino RatioSR = Sharpe RatioUS = United StatesUSA = United States of AmericaVaR = Value-at-RiskVT = Volatility TimingVW = Value-Weighted

1. Introduction 1

1. Introduction

In 1952, Harry Markowitz published his work Portfolio Selection and arguably set thefoundation of what is nowadays known as Modern Portfolio Theory. Regarding a set ofrisky assets, Markowitz (1952) rationalizes an investment behavior that favors expectedreturn and dismisses risk measured in terms of variance of return. Since returns typicallytend to increase with risk, this constitutes a fundamental trade-off which, however, canbe addressed by a rational and risk-averse investor via fixing a level of variance of returnwhile maximizing expected return. Although theoretically appealing, there is no guaran-tee that following this proposed investment behavior indeed results in desired outcomeswhen tested in a real data environment. A natural choice of a desired outcome is theoutperformance of an optimized portfolio strategy relative to the naive diversification orsimply 1/N approach. The latter describes an investment schedule which invests equalshares of wealth in all available risky assets of which the number is assumed to be N.

In this spirit, this analysis conducts an out-of-sample horse race between the naive di-versification approach and 15 optimized portfolio strategies based on the mean-varianceframework. By doing so, several contributions to the existing related portfolio optimiza-tion literature are made. First, similar studies such as DeMiguel et al. (2009b) are up-dated by respecting well-established mean-variance model extensions as well as rathernewly proposed approaches resulting in a comprehensive selection of optimized portfoliostrategies. Second, the data sets are not exclusively restricted to the United States (US)but also account for European equity data and hence allow for a comparison betweenUS and European specific results. Third, the scope of performance evaluation criteria isextended and contains an additional number of four reward-to-risk ratios alongside thestandard Sharpe ratio and certainty equivalent return. Besides, the test statistic introducedby Ledoit and Wolf (2008) to decide on the pairwise difference between the Sharpe ratiosof two different strategies is adjusted so as to be also valid in case of certainty equivalentreturns. This facilitates the identification of statistically significant results and explicitlyaccounts for nonnormally distributed return data. The results of this analysis are benefi-cial not only for private investors but also for professional practitioners who strive for amore favorable return-to-risk trade-off and have an interest in answering the question ofwhether optimized portfolios can beat 1/N.

The main finding of this analysis can be summarized in the fact that not a single op-timized portfolio strategy achieves to consistently outperform the naive diversificationbenchmark in a statistically significant manner. However, performances across differentdata sets are very heterogeneous. In case of two US equity data sets formed on bivariatesorts of market equity and the book-to-market equity ratio, unconstrained strategies re-lated to the sample-based mean-variance and minimum-variance strategy exhibit superiorperformances. In contrast, timing strategies only attain consistently good results in thecontext of US equity data classified according to US industries. The European data setsare generally short of any outperforming optimized strategy. Eventually, the performance


behavior appears to be very sensitive to respective data sets. The relative performancerankings of the strategies under consideration, however, are independent of the perfor-mance evaluation criteria, i.e. the Sharpe ratio is in general an adequate choice althoughrelying on the assumption of normally distributed return data.

This analysis is organized as follows. Section 2 reviews relevant literature and presentsimportant empirical results. Section 3 describes the data. Section 4 introduces the opti-mized portfolio strategies and Section 5 lists the performance evaluation criteria. Resultsare presented and discussed in Section 6. Section 7 contains robustness checks and Sec-tion 8 concludes.

2. Literature Review

The past few years of portfolio optimization research can be characterized by a seriesof publications conducting horse races between different competing optimized portfoliostrategies. In this context, the work by DeMiguel et al. (2009b) is of particular interestand also serves as the basis on which this analysis is build.

DeMiguel et al. (2009b) compare the performance of 12 different optimized portfoliostrategies relative to the naive diversification benchmark with respect to the out-of-sampleSharpe ratio, certainty equivalent return, and turnover. The underlying data are composedof monthly excess stock returns for mainly US equities and cover the period from July1963 to November 2004. By applying a rolling-window estimation approach, DeMiguelet al. (2009b) come to the somewhat surprising conclusion that none of the strategies un-der consideration achieves to consistently outperform the naive diversification approach.This result is valid throughout all performance criteria. DeMiguel et al. (2009b) attributethe bad performance of the optimized strategies to the characteristic occurrence of esti-mation errors when estimating expected returns and variance-covariance matrices in theprocess of building these optimal strategies.

Tu and Zhou (2011) subsequently introduce four novel optimized portfolio strategiesthat are optimized combinations of already existing approaches. In particular, the naivediversification approach is combined with the sample-based mean-variance strategy aswell as with the strategies suggested by Jorion (1986), MacKinlay and Pástor (2000), andKan and Zhou (2007), respectively. The general setting and estimation methodology isvery similar to DeMiguel et al. (2009b) which ensures a certain degree of comparability.Tu and Zhou (2011) show evidence of an outperformance of the optimally combinedportfolio strategies relative to their uncombined counterparts in terms of Sharpe ratios andcertainty equivalent returns. Some of the newly proposed optimized portfolio strategieseven achieve to outperform naive diversification. This is especially true in case of largesample sizes. These findings are promising and rationalize the use of a subset of theirproposed models also in this analysis.


Pflug et al. (2012) focus on the two levels of uncertainty prevailing in the context ofportfolio optimization. First, the actual realizations of the asset returns are uncertain andsecond, and even more critical, the data generating process itself is not known exactly.The higher the degree of this model uncertainty the more the theoretically optimal port-folio strategy resembles the naive diversification approach. On the basis of different riskmeasures, Pflug et al. (2012) subsequently demonstrate that naive diversification is evenoptimal in case of a risk-averse investor who cares about both levels of uncertainty incombination with a very limited or even nonexistent knowledge of the data generatingprocess. As a result, the empirical success of the naive diversification approach cannotonly be attributed to general estimation inaccuracies when estimating moments of an as-sumed data generating distribution but also to wrong inferences about the data generatingprocess in the first place. Following this argument, focus should be either put on the im-provement of the estimation precision of already existing models or the introduction ofnew optimized portfolio strategies including a justified modeling of the data generatingprocess.

An interesting remark is made by Kirby and Ostdiek (2012) who argue that the de-sign of the comparisons between the different portfolio strategies typically favors naivediversification. The general assumption of a representative mean-variance investor leadsto optimized portfolio strategies that strive for high expected returns. This, however, canresult in extreme portfolio weight positions that take advantage of even minor asset returndifferences. In case of an imprecise estimation of these return rates, the resulting portfoliocan exhibit poor out-of-sample performance. Hence, Kirby and Ostdiek (2012) proposethe idea of imposing a target return on the optimized portfolio strategies equal to the out-of-sample return obtained by the naive diversification approach. This indeed improvesthe performance of the optimized portfolio strategies relative to naive diversification anddampens the portfolio weights to less extreme positions. In addition, Kirby and Ostdiek(2012) introduce a volatility and reward-to-risk timing strategy which achieve superiorresults relative to the naive diversification benchmark for US equity data covering the pe-riod from July 1963 to December 2008. The performance of these two newly introducedstrategies is encouraging as it suggests that optimized portfolio strategies can actuallyoutperform naive diversification irrespective of the reliance on sufficiently long samplesizes.

An extension of the work by DeMiguel et al. (2009b) to a broader set of international as-set classes including stocks, bonds, and commodities is given by Jacobs et al. (2014). Forthe time period from February 1973 to December 2008, a total of eleven mean-variancebased optimized portfolio strategies are compared to the naive diversification approach.As a first result, Jacobs et al. (2014) confirm previous findings that hardly any optimizedportfolio strategy consistently outperforms naive diversification with respect to equitydata. Second, these results transmit to the other asset classes under consideration forwhich naive diversification also seems to be the most successful approach in achieving

3. Data 4

high out-of-sample Sharpe ratios.In the same spirit, Bessler et al. (2017) examine the period from January 1993 to

December 2011 with respect to an international portfolio consisting of stocks, bonds,and commodity indices and compare the naive diversification approach to the optimizedsample-based mean-variance, minimum-variance, and Bayes-Stein strategy. In accor-dance with DeMiguel et al. (2009b), the Sharpe ratios belonging to the optimized strate-gies do not significantly outperform naive diversification. Additionally, the Omega ratioas well as the maximum drawdown are included as further reward-to-risk measures andaccording to these criteria the obtained results are suggestive of a superior performanceregarding the optimized portfolio strategies.

This analysis aims at continuing this series of portfolio optimization research by con-ducting a horse between optimized portfolio strategies that incorporates the main insightsof previous empirical work with respect to both the selection of optimized strategies aswell as performance evaluation criteria.

3. Data

This analysis is based on a total of six data sets containing monthly excess returns overa risk-free rate defined as the one-month US Treasury bill rate. There are four samplesbased on US data and two samples referring to European data. The returns of the USdata sets are a combination of New York Stock Exchange (NYSE), American Stock Ex-change (AMEX), and National Association of Securities Dealers Automated Quotations(NASDAQ) stocks and are value-weighted portfolios either formed based on bivariatesorts on market equity and the book-to-market equity ratio or ordered by industry affil-iation. The US data cover the time period from January 1970 to December 2017. Thechoice of the time period is determined by data availability. The US industry data areonly complete as of the year 1970. The return series for the other US samples originallyinclude more observations but are adjusted for the same time period as the US industrydata to ensure comparability. Independent of the actual data availability, all data sets arebalanced such that the time period always starts in January and ends in December. Thisis done to guarantee an equal distribution of months to mitigate calendar effects such asthe well-known January effect. As a result, the US portfolios each contain a total of 576months of excess return data representing 48 years of excess returns, respectively. Thereturns of the European data sets are based on data provided by Morgan Stanley CapitalInternational (MSCI) and Bloomberg and are value-weighted portfolios formed based onbivariate sorts on market equity and the book-to-market equity ratio. The time periodranges from January 1991 to December 2017 and thus includes 324 observations alto-gether corresponding to 27 years of monthly excess return data. All data belong to thesingle-asset class of stocks and are drawn from Kenneth R. French’s data library1. The

1http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html

http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html

3. Data 5

Table 3.1: Overview Data Sets. This table lists a total of six data sets that are included in this analysisall of which contain monthly excess returns over a risk-free rate defined as the one-month US Treasury billrate. The first four samples refer to US data while the remaining two samples are based on European data.The abbreviations are introduced to refer to the specific data sets in the text. The variable N indicates thetotal number of available risky assets included in each data set. The time period indicates the duration ofthe data collection and the last column of the table lists the underlying number of observations included ineach data set. The data are obtained from Kenneth R. French’s website http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html. A more detailed description of the data andthe process of creating the different portfolios can be found in Appendix A.

# Data Set Abbreviation N Time Period Obs.

United States of America (USA)1 Six US portfolios formed on market equity

and the book-to-market equity ratioPF6 USA 6 01/1970 - 12/2017 576

2 Twenty-five US portfolios formed on mar-ket equity and the book-to-market equityratio

PF25 USA 25 01/1970 - 12/2017 576

3 Ten US industry portfolios IND10 USA 10 01/1970 - 12/2017 5764 Forty-eight US industry portfolios IND48 USA 48 01/1970 - 12/2017 576

Europe (EU)5 Six European portfolios formed on market

equity and the book-to-market equity ratioPF6 EU 6 01/1991 - 12/2017 324

6 Twenty-five European portfolios formedon market equity and the book-to-marketequity ratio

PF25 EU 25 01/1991 - 12/2017 324

data sets employed in this analysis are very similar to the samples typically used in therelated empirical portfolio optimization literature and therefore a high degree of compa-rability with respect to the obtained results is guaranteed. Table 3.1 presents an overviewof the data considered in this analysis.

The remainder of this section is dedicated to the description of PF6 USA and PF6 EUas well as their respective value-weighted excess market return (MKT) series to obtaina deeper understanding of the inherent characteristics of the data sets underlying thisanalysis. In case of the PF6 portfolios, the total stock data are divided into a small and abig group with respect to a specific level of market equity of which each group is againsubdivided into either a low, medium, or high group with respect to a specific level ofbook-to-market equity ratio. A more detailed description of the data and the process ofcreating the different portfolios can be found in Appendix A. Table 3.2 exhibits descriptivestatistics for the US and European MKT series and PF6 portfolios, respectively. Severalinteresting facts appear when examining Table 3.2. Focusing on the MKT series of theUSA and Europe, a first observation refers to the mean of the US MKT series whichis substantially higher relative to that of Europe. In addition, the European market ischaracterized by a higher standard deviation and more extreme values as indicated bythe corresponding minimum and maximum values. These facts summarize in a EuropeanMKT Sharpe ratio of approximately 0.12 which is lower than the respective value of 0.17for the USA. Based on this reward-to-risk ratio, the descriptive statistics are suggestiveof an outperformance of the European market by the US market. A second observation is



3. Data 6

Table 3.2: Descriptive Statistics for the US and European MKT Series and PF6 Portfolios. This tablelists the MKT series and the six portfolios for the US and Europe, i.e. PF6 USA and PF6 EU, respectively.The six portfolios are constructed by means of bivariate sorts of the total stock data on market equity withtwo groups, small and big, and on book-to-market equity ratio with three groups, low, medium, and high.The US portfolios are truncated so as to match the time period of the European portfolio. Hence, thedata span from January 1991 to December 2017 containing 324 observations of monthly excess returnsrepresenting 27 years. This table displays the mean, standard deviation, sample-bias corrected skewness,sample-bias corrected kurtosis, minimum value, maximum value, and Sharpe ratio of each return series.

USA Small BigStatistics MKT Low Medium High Low Medium High

Mean 0.0072 0.0090 0.0125 0.0135 0.0097 0.0094 0.0102Standard Deviation 0.0417 0.0657 0.0499 0.0531 0.0417 0.0410 0.0499Skewness −0.7082 −0.2178 −0.4815 −0.6509 −0.4712 −0.7527 −0.8022Kurtosis 4.5338 4.5721 4.6741 4.8114 4.0160 5.6602 5.7157Minimum −0.1723 −0.2447 −0.1919 −0.2049 −0.1499 −0.1797 −0.2227Maximum 0.1135 0.2838 0.1663 0.1729 0.1444 0.1239 0.1766Sharpe Ratio 0.1727 0.1370 0.2505 0.2542 0.2326 0.2293 0.2044Europe Small BigStatistics MKT Low Medium High Low Medium High

Mean 0.0057 0.0056 0.0082 0.0104 0.0065 0.0086 0.0086Standard Deviation 0.0483 0.0530 0.0487 0.0501 0.0457 0.0495 0.0572Skewness −0.5951 −0.8329 −0.9231 −0.7339 −0.5482 −0.6478 −0.4904Kurtosis 4.8052 5.9839 6.8226 6.4994 4.6034 4.7857 4.8253Minimum −0.2202 −0.2592 −0.2654 −0.2695 −0.1900 −0.2073 −0.2473Maximum 0.1367 0.1674 0.1515 0.1648 0.1331 0.1411 0.2127Sharpe Ratio 0.1180 0.1057 0.1684 0.2076 0.1422 0.1737 0.1503

that throughout all portfolios the minimum and maximum values are relatively extreme inrelation to the corresponding means and standard deviations. This behavior is indicativeof outliers. Also, all portfolios show a negative skewness as well as a level of kurtosis thatexceeds the reference level of a standard normal distribution of three. These observationscast a first doubt on the general assumption prevailing in the empirical literature of havingnormally distributed asset returns.

A graphical representation of some distributional characteristics of the US and Euro-pean MKT series and PF6 portfolios can be obtained by boxplots as shown in Fig. 3.1.The lengths of the boxes for the European sample appear to be more homogenous relativeto the US sample suggesting that the European portfolios share more similar distributionalcharacteristics. Again, the number of outliers in all boxplots speak in favor of return datawhich are not normally distributed.

Table 3.3 yields correlations of the US MKT series and PF6 portfolios. All correlationsare highly positive and even the lowest correlation coefficient between the US B-M andS-L portfolios still amounts to 0.62. In addition, the correlations between the US andEuropean portfolios are highly positive ranging from 0.62 for the S-H portfolio to 0.79for the MKT series. Facing an investment universe that only consists of highly correlatedassets limits the potential benefits of diversification and might result in lower levels of

3. Data 7

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

B-H

B-M

B-L

S-H

S-M

S-L

MKT

Monthly Excess Returns

MK

TSe

ries

and

PF6

Port

folio

s

Boxplots for US MKT and PF6

(a) Boxplots for the US MKT series and PF6 portfo-lios. The US portfolios are truncated so as to matchthe time period of the European portfolios. Hence,the data spans from January 1991 to December 2017containing 324 observations of monthly excess re-turns representing 27 years.

−0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25

B-H

B-M

B-L

S-H

S-M

S-L

MKT


MK

TSe

ries

and

PF6

Port

folio

s

Boxplots for European MKT and PF6

(b) Boxplots for the European MKT series andPF6 portfolios. The data spans from January 1991to December 2017 containing 324 observations ofmonthly excess returns representing 27 years.

Figure 3.1: Boxplots for the US and European MKT Series and PF6 Portfolios. This figure showsboxplots for the US and European MKT series and PF6 portfolios, respectively. The vertical line in eachbox denotes the median of the distribution. The left and right bounds of the box indicate the 25th and75th percentiles, respectively. The whiskers cover the range of the observations that are still within the 1.5interquartile range of the lower and upper quartile. Outliers are denoted by a dot.

reward-to-risk ratios due to higher levels of risk.Fig. 3.2 displays the monthly cumulative MKT series for the USA and Europe. The US

sample starts in January 1970, whereas the European sample only starts in January 1991due to limited data availability. Both series are upward sloping and show a clear positivetrend. Moreover, both return patterns resemble each other, however, the US cumulativereturns are higher relative to the European returns for most of the underlying time period.Several major drops in the observed return patterns can be attributed to correspondingeconomic crises such as the dot-com bubble in the beginning of the 21st century as well

Table 3.3: Correlation Table for the US and European MKT Series and PF6 Portfolios. This table listscorrelations for the US and European MKT series and PF6 portfolios, respectively. The US portfolios aretruncated so as to match the time period of the European portfolios. Hence, the data span from January 1991to December 2017 containing 324 observations of monthly excess returns representing 27 years. This tablecontains several different correlations. In fact, columns 2 (MKT) to 7 (B-H) span two correlation matrices atthe same time. The lower left triangular matrix containing plain numbers represents the correlation matrixof the US sample. The upper right triangular matrix containing italic numbers represents the correlationmatrix of the European sample. The last column USA/EU yields the correlations of the portfolios betweenthe USA and Europe.

Factors MKT S-L S-M S-H B-L B-M B-H USA / EU

MKT 1.00 0.89 0.92 0.89 0.95 0.99 0.96 0.79S-L 0.83 1.00 0.95 0.88 0.85 0.85 0.81 0.66S-M 0.85 0.91 1.00 0.96 0.84 0.89 0.87 0.66S-H 0.81 0.84 0.96 1.00 0.78 0.87 0.89 0.62B-L 0.97 0.78 0.75 0.69 1.00 0.93 0.85 0.77B-M 0.89 0.62 0.77 0.77 0.82 1.00 0.94 0.78B-H 0.85 0.63 0.79 0.82 0.76 0.89 1.00 0.72

3. Data 8

1970 1980 1990 2000 2010 2020−0.5

0

0.5

1

1.5

2

2.5

3

3.5

DateM

onth

lyC

umul

ativ

eM

KT

Seri

es

Cumulative US and European MKT Series

MKT USAMKT Europe

Figure 3.2: Monthly Cumulative MKT Series for the US and Europe. This figure shows the monthlycumulative MKT series for the US and Europe. The US sample covers the period from January 1970 toDecember 2017 containing 576 observations of monthly excess returns representing 48 years. The Europeansample covers the period from January 1991 to December 2017 containing 324 observations of monthlyexcess returns representing 27 years.

as the global financial crisis in 2007 and 2008.Since the actual distribution of the underlying return data critically influences the choice

of appropriate statistical procedures, several different tests are performed. Tests for nor-mality with respect to a specific sample distribution can rely either on a graphical ap-proach or more formally utilizing actual statistical tests. Fig. 3.3 depicts the histogram,a fitted normal distribution, and a kernel density estimate for the US and the EuropeanMKT series. A visual inspection of the fitted normal distribution relative to the kerneldensity estimate does not yield a huge discrepancy neither for the US nor for the Euro-pean sample. Only the European return distribution exhibits slightly fat tails, especiallyon the left-hand side of the kernel density estimate. These fat tails are characteristic of re-

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.20

2

4

6

8

10

12


Den

sity

Histogram, Normal Density, and Kernel Density Estimate

for the US MKT Series

HistogramNormal DensityKernel DensityEstimate

(a) Histogram, normal density, and kernel density es-timate for the US MKT series based on a total of576 monthly excess return observations from Jan-uary 1970 to December 2017. The kernel functionis the Epanechnikov with a bandwidth of 0.0124.

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.150

2

4

6

8

10

12


Den

sity

Histogram, Normal Density, and Kernel Density Estimate

for the European MKT Series

HistogramNormal DensityKernel DensityEstimate

(b) Histogram, normal density, and kernel densityestimate for the European MKT series based ona total of 324 monthly excess return observationsfrom January 1991 to December 2017. The kernelfunction is the Epanechnikov with a bandwidth of0.0147.

Figure 3.3: Histogram, Normal Density, and Kernel Density Estimate for the US and European MKTSeries This figure shows the histogram, normal density, and kernel density estimate for the US and Euro-pean MKT series. The kernel function is the Epanechnikov with unbounded support.

3. Data 9

−4 −3 −2 −1 0 1 2 3 4−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Theoretical Quantiles of the Normal Distribution

US

MK

TQ

uant

iles

Quantile-Quantile Plot for the US MKT Series

(a) Quantile-quantile plot for the US MKT seriescovering the period from January 1970 to December2017 containing 576 observations of monthly excessreturns representing 48 years.

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Theoretical Quantiles of the Normal Distribution

Eur

opea

nM

KT

Qua

ntile

s

Quantile-Quantile Plot for the European MKT Series

(b) Quantile-quantile plot for the European MKTseries covering the period from January 1991 toDecember 2017 containing 324 observations ofmonthly excess returns representing 27 years.

Figure 3.4: Quantile-Quantile Plot for the US and European MKT Series. This figure shows thequantile-quantile plots for the US and European MKT series. The straight line represents perfectly nor-mally distributed data and serves as a benchmark for comparison with the actual MKT series.

turn series and the reason why return data is often better described by a leptokurtic ratherthan a normal distribution.

Quantile-quantile plots such as given in Fig. 3.4 allow for a comparison of two distri-butions. To be precise, the actual return data of the US and European MKT samples arecompared to a corresponding normally distributed sample represented by the straight line,respectively. Clearly, both ends of both return series do not fall along the straight line,hence indicating that the MKT series are not normally distributed.

The assumption of normally distributed return data is also examined by employingtwo statistical tests. Both the Jarque-Bera test as well as the Lilliefors test are two-sidedgoodness-of-fit tests of whether the sample data stem from a normal distribution with un-known and hence estimated mean and variance. The Jarque-Bera test compares the pre-

Table 3.4: Normality and Stationarity Tests for the US and European MKT Series. This table lists theJarque-Bera and the Lilliefors test for normality as well as the augmented Dickey-Fuller test for stationarityof a time series. The normality tests challenge the null hypothesis of having a normally distributed sampleto the alternative hypothesis of the sample data being nonnormal. The test for stationarity challenges thenull hypothesis of having a time series characterized by MKTt = MKTt−1 + εt, i.e. the time series contains aunit root and is nonstationary, to the alternative hypothesis of the time series being stationary and followingan autoregressive (AR) process of order one, AR(1), i.e. MKTt = φMKTt−1 + εt with φ < 1 and whitenoise εt without any drift or deterministic time trend. The tests are applied to the US and European monthlyMKT series covering the period January 1970 to December 2017 and January 1991 to December 2017,respectively. This table displays the test statistics of each test and the corresponding p-values.

USA EuropeStatistic p-Value Statistic p-Value

Tests for NormalityJarque-Bera 123.2157 0.0010 60.7163 0.0010Lilliefors 0.0553 0.0010 0.0663 0.0016Test for StationarityAugmented Dickey-Fuller −21.9360 0.0010 −15.9355 0.0010

4. Optimized Portfolio Strategies and 1/N 10

dicted skewness and kurtosis of the normal distribution to the actual data sample, whereasthe Lilliefors test compares the empirical distribution function of the data sample with thecumulative distribution function of the predicted normal distribution. As shown in Ta-ble 3.4 and already recommended by the visual inspection of the MKT series, both testsreject the null hypothesis that the sample distribution is normal at the 5% significancelevel. This is true for the US as well as for the European MKT series. Table 3.4 alsopresents the test statistic and the corresponding p-value of the augmented Dickey-Fullertest for stationarity. The test rejects the null hypothesis at the 5% significance level sug-gesting a stationary MKT series for both the US and Europe. This result also implies thatthe data generating parameters such as the mean and variance do not change over time.

To sum up, the above findings are suggestive of an outperformance of the Europeanmarket by the US market based on both a higher mean return and smaller standard devia-tion. In addition, the specific process of creating the portfolios considered in this analysisresults in high positive correlations of assets within the different data sets. All return se-ries exhibit the common characteristics of having a negative skewness and high kurtosis.This speaks in favor of frequent small return gains combined with the possibility of fewrather extreme return losses. In fact, statistical tests reject the null hypothesis of havingnormally distributed return series, an information that needs to be accounted for to ensurereliable statistical inferences.

4. Optimized Portfolio Strategies and 1/N

This section presents a brief overview of different approaches on how to create optimizedportfolios when facing estimation risk as is done in the related empirical literature. Sub-sequently, the optimized portfolio strategies employed in this analysis are introduced.However, to begin with, some notation and the general setting are defined.

4.1. Mean-Variance Framework

Return and risk measured in terms of variance of return are the key characteristics whenconsidering assets and portfolios of assets. Due to their pivotal role with respect to port-folio optimization, it is crucial to understand the basic mechanisms at work when assetswith different properties are combined to create a portfolio of assets. Let r = (r1, . . . , rN)T

denote the vector of uncertain excess returns for an investment universe consisting of N

risky assets over the risk-free rate r f , i.e. r = R − r f 1N with the complementary vector ofnonexcess returns R ∈ RN×1 and 1N is a vector of ones with dimension N. The expectedexcess returns of the assets µ(r1, . . . , rN) can be stated as

µ =(E(r1), . . . , E(rN)

)T. (4.1)


The symmetric N × N variance-covariance matrix of the returns Σ(r1, . . . , rN) is given by

Σ = E((r − µ)(r − µ)T )

=

V(r1) Cov(r1, r2) . . . Cov(r1, rN)

Cov(r2, r1) V(r2) . . . Cov(r2, rN)...

.... . .

...

Cov(rN , r1) Cov(rN , r2) . . . V(rN)

=

σ2

1 σ12 . . . σ1N

σ21 σ22 . . . σ2N

......

. . ....

σN1 σN2 . . . σ2N

(4.2)

with the variance of asset i denoted by σ2i and σi j = σiσ jρi j is the covariance between

asset i and j with correlation coefficient ρi j. A portfolio can be completely characterizedby specifying the proportion that each available risky asset takes in the portfolio. Thisinformation can be stored in x = (x1, . . . , xN)T with the individual weighting of asset i

denoted by xi such that x f = 1 − 1TN x is the investment in the risk-free asset and the

representative investor is fully invested, i.e. the weights sum up to one. Consequently, thereturn of a portfolio in excess of the risk-free rate rp(x1, . . . , xN) takes the form

rp =

N∑i=1

xiri = xT r. (4.3)

The expected excess return of a portfolio µp(x1, . . . , xN) consisting of N different assetscan be stated as

µp = E(rp) =

N∑i=1

xiE(ri) = xTµ. (4.4)

The corresponding variance of a portfolio σ2p(x1, . . . , xN) can be specified as

σ2p =

N∑i=1

x2iσ

2i +

N∑i=1

N∑j=1j,i

xix jσiσ jρi j = xT Σx, (4.5)

where Σ is expected to be positive definite, i.e. xT Σx > 0, x , 0, to guarantee that theassets under consideration are not redundant, i.e. are linearly independent with respectto their return pattern. Although the existence of such an asset would not alter the resultof the underlying optimization problem the invertibility of the variance-covariance matrixneeds to be secured due to computational issues.

For the representative investor to behave in accordance with the expected utility hypoth-esis at least one of the two assumptions have to be fulfilled. Either the investor is subject to


have quadratic utility, although other utilities can be evaluated utilizing a second-order ap-proximation, or the distributions of asset returns are jointly normally distributed, i.e. canbe completely characterized by only the first two moments, while the individual returnsare all independent and identically distributed, respectively. Pennacchi (2008) provides ageneral proof. Quadratic utility implies an increasing relative and absolute risk aversionwith increasing investor’s wealth, a fact that seems to contradict realistic behavior as ar-gued in Cohn et al. (1975) and Morin and Suarez (1983). Normally distributed returnsrefer to the class of elliptical distributions such as the normal Gaussian distribution. How-ever, empirically returns rather follow a leptokurtic distribution that cannot entirely bedescribed only by its mean and variance in accordance to the findings of Section 3. Also,investors aiming at maximizing expected utility not necessarily only care about the firsttwo moments of the underlying distribution but might have also preferences for highermoments. Despite these potential limitations and assumptions, the beauty of a model isnot based on the exact replication of the complex world in all detail but rather on ab-straction and simplification while still maintaining the actual purpose of a model that isproviding meaningful insights.

In this spirit and given a time period of T observations, the representative investorchooses a composition xt at every period t, t = 1, . . . ,T , of the available N risky assets tocreate a portfolio so as to maximize expected utility, E(u), i.e.

maxxt

E(u(xt)

)= xT

t µt −γ

2xT

t Σtxt. (4.6)

The expected utility is an increasing function in the portfolio’s expected excess return anddecreasing in the portfolio’s variance with the investor’s specific risk-aversion parameterγ representing the strength of this fundamental trade-off between expected return and risk.It is not necessary to explicitly constrain the weights to sum to one as this condition isalready implicitly incorporated in the optimization problem due to the fact that excessreturns rather than nonexcess returns are considered, see Appendix B. This optimizationproblem has a closed-form solution that can be obtained by taking the first derivative ofEq. (4.6) with respect to xt and takes the form

xt =1γ

Σ−1t µt. (4.7)

In general, this solution implies an allocation to the unique risky portfolio as well as to therisk-free asset with a larger share invested in the risk-free asset the higher the investor’srisk-aversion parameter. Finally, the adjusted weights

ωt =xt∣∣∣1TN xt

∣∣∣ =

1γΣ−1

t µt∣∣∣∣1TN

1γΣ−1

t µt

∣∣∣∣ =Σ−1

t µt∣∣∣1TNΣ−1

t µt

∣∣∣ (4.8)

yield the relative amount allocated to the risky portfolio at date t. The rescaling by the


absolute value guarantees a proportional investment between the risky portfolio and therisk-free asset across the different strategies such that performance differentials do notdepend on an unequal investment share with respect to the risk-free asset.

4.2. Estimation Risk

The optimization problem in Eq. (4.6) crucially depends on the estimation of expectedreturns, variances, and covariances. The number of parameters one would need to esti-mate to compute the optimal mean-variance portfolio when considering 100 risky assetsalready amounts to 5,1502. Apart from the fact that the estimation of a large number ofquantities can be rather time consuming and a computational challenging task, there isalso a certain level of risk inherent in every estimation. In fact, there is no guaranteethat the obtained estimates coincide with the actual true underlying values. Already Mer-ton (1980) shows evidence that the estimation of the variance-covariance matrix is muchmore precise relative to the estimation of the corresponding expected returns. Chopra andZiemba (1993) further specify that the most critical estimations with respect to estima-tion accuracy are the expected returns, followed by the variance estimates and lastly theestimates of the covariances. The difficulty in estimating expected returns has its originin the typically observed relatively small average return values in combination with rel-atively high return volatilities such that only a very long time series of observations canguarantee an accurate estimation. Broadie (1993) calculates that 26 years of monthly re-turn data are needed to correctly distinguish with a probability of 90% between two assetswith different expected monthly returns of 1 and 1.5%, respectively. This result assumesnormally distributed returns, a common standard deviation of 7%, and a correlation co-efficient of 0.5. This example nicely illustrates how difficult it is to obtain precise returnestimates or rather how return estimates critically depend on the number of observationsavailable for estimation. In contrast, less than five years of monthly data are required todistinguish with a probability of 90% between two assets with different standard devia-tions of 6 and 7%, respectively. This computation is based on normally distributed returnswith a common mean of 1% and a correlation coefficient of 0.5, see Broadie (1993). Still,having a large data set also introduces new challenges to the estimation process since theprobability of the time series being nonstationary, for example, increases with the lengthof the data set as argued in Jobson and Korkie (1980). This constitutes a trade-off betweenestimation precision and estimation validity that is omnipresent in the context of out-of-sample portfolio optimization. Over the past decades, the related empirical literature hasdeveloped different econometric approaches on how to mitigate this estimation risk.

2For a set of N ∈ N risky assets, N returns ri, N variances σ2i , and N(N − 1)/2 corresponding covariances

σi j, i = 1, . . . ,N, i , j, need to be estimated resulting in a total of 2N + N(N − 1)/2 parameters.Additional parameters needed are the risk-free rate r f and the risk-aversion parameter γ.


4.3. Approaches to Fight Estimation Risk

The Bayesian approach starts from the premise that the parameters of the data generatingprocess are not known to the representative investor. As a result, the investor is facedwith the task of incorporating an educated guess about the data generating process intothe expected utility maximization problem. These a priori inferences, or priors, representthe investor’s beliefs about the parameters of the data generating process before beingpresented with evidence. The class of priors is commonly divided into uninformativeor diffuse priors and informative priors. The former such as proposed in Barry (1974),Klein and Bawa (1976), and Brown (1979) do not systematically incorporate any specificinformation about the parameters of the return distribution. Since the Bayesian approachexplicitly accounts for potential estimation errors, the already risky assets become evenmore risky. The risk of the risk-free asset, however, remains zero by assumption. Incomparison to the mean-variance plug-in approach, the Bayesian solution hence typicallyinvests relatively less in the risky assets and more in the risk-free asset. Nevertheless, theresults obtained from these general diffuse priors are very similar to the mean-varianceapproach and even coincide in case of an infinitely long time series of observations, seeBrandt (2009). In contrast, informative priors do incorporate specific information. Thearguably most well-known approach is based on Black and Litterman (1992).

First introduced by Stein (1956) and then extended by James and Stein (1961), theshrinkage technique also aims at reducing potential estimation errors. The basic idea ofshrinkage, nicely illustrated in Efron and Morris (1977), is that the quality of the estimateof the expected returns should increase if the sample mean is shrinked toward a commonvalue or grand mean, i.e. the mean of the means across all variables. In particular, theshrunked mean is supposed to be less likely affected by extreme observations relative tothe sample mean. This approach can be understood as a special case of the Bayesian tech-nique in which the prior is the shrinkage target. The resulting reduction in the estimate’svariance is assumed to be more beneficial than the harm that occurs by shrinking and thusbiasing the estimate. Shrinkage can thus be seen as a trade-off between either incurringa bias or having low levels of variance. In general, the shrinkage factor is an increasingfunction in the number of assets and a decreasing function in the number of time periodsconsidered. Jobson and Korkie (1980), Jorion (1985), and Jorion (1986) focus on shrink-ing expected returns, whereas Frost and Savarino (1986), Ledoit and Wolf (2003), andLedoit and Wolf (2004) extend the traditional approach and apply the shrinking techniqueon variances and covariances. Kourtis et al. (2012) avoid the detour and directly shrinkthe inverse of the variance-covariance matrix. Moreover, shrinkage is not only limitedto the moments of the asset return distribution but can also be applied to optimal portfo-lio weights. Applications can be found in Golosnoy and Okhrin (2007) and Frahm andMemmel (2010).

A further specification of a prior is given by factor models. Factor models try to fa-cilitate the estimation of expected returns by identifying a limited number of variables


able to explain the observed variation in asset return cross-sections or time series. TheCapital Asset Pricing Model (CAPM) as suggested by Sharpe (1964), Lintner (1965), andMossin (1966) represents a special case of a single-factor model and links the expectedasset return to the expected excess market return. The number of parameters one wouldneed to estimate to compute the optimal mean-variance portfolio when considering 100risky assets decreases from 5,150 based on the mean-variance plug-in approach to 3023.This reduction in the number of parameters to be estimated illustrates the advantages offactor models with respect to the estimation procedure. Applications of this approach canbe found in Pástor (2000) and Pástor and Stambaugh (2000). However, the workhorse ofthe empirical factor model literature is the Fama-French three-factor model as proposedin Fama and French (1993) including its extensions to a four-factor model as in Carhart(1997) and a five-factor model as in Fama and French (2015).

Additional strands consider different specific portfolio restrictions and hence are sup-posed to model real-world trading in a more realistic fashion. By doing so, short saleconstraints play a key role. In this context, Jorion (1992) proposes an interesting thoughtexperiment. He considers a world with two assets in which asset A and B are assumedto have an estimated average return of 10.1 and 9.9%, respectively, although both assetshave a true return of 10%. A short sale restricted investor would choose to invest mainlyin asset A based on its higher return. Without short-sale constraints, however, the investorwill heavily buy asset A and short asset B to take advantage of the return differential of0.2%. Since the resulting profit of the portfolio is increasing with the long position inasset A and the short position in asset B, the optimization process can result in very ex-treme investment positions. In response to that finding, constraining the weights to benonnegative not only mitigates the chance of adopting extreme portfolio weights due toestimation error but also leads to more realistic outcomes. DeMiguel et al. (2009a) extendthis approach by focusing on the minimum-variance portfolio while constraining differentnorms of the portfolio weight vector to be smaller than a given threshold. Furthermore, itis also possible to restrict short sales only to specific single assets or to allow short salesin general but only up to a limited degree. Other real-world examples of restrictions thatinvestors might face are specific constitutional regulations or policy constraints that onlyallow the investment in socially responsible assets such as given by companies that do notviolate human rights, support corruption, or negatively affect the environment.

4.4. Portfolio Strategies

The selection of models included in this analysis aims at fulfilling two objectives. First,past empirical literature is respected by including already well-established optimized port-folio strategies such as the sample-based mean-variance and the Bayes-Stein strategy.

3There are only 3N + 2 parameters that need to be estimated namely N returns, N betas, N variances, theexpected market return, and the variance of the market return. Additional parameters needed are therisk-free rate r f and the risk-aversion parameter γ.


Table 4.1: Overview Portfolio Strategies. This table lists a total of 16 portfolio strategies that are includedin this analysis. The first model is the naive diversification approach that also serves as the benchmarkmodel on which the performances of the remaining 15 optimized portfolio strategies are measured against.The abbreviations are introduced to refer to the specific model in the text. The last column of the tablelists the number of estimated parameters needed to implement a specific model under the assumption ofan investment universe consisting of N different risky assets. This number excludes the estimation of therisk-free rate r f and the risk aversion parameter γ.

# Portfolio Strategy Abbreviation Number of Estimations

Naive Diversification1 Equally-weighted EW 0

Sample-Based Mean Variance

2 Sample based mean-variance MV (N2 + 3N)/2Moment Restrictions

3 Minimum-variance MinV (N2 + N)/24 Value-weighted VW 0

5 MacKinlay-Pástor MP (N2 + 3N)/2Bayesian Approach

6 Bayes-Stein BS (N2 + 3N)/2Portfolio Weight Constraints

7 Mean-variance short sale constraint MVsc (N2 + 3N)/2

8 Minimum-variance short sale constraint MinVsc (N2 + N)/2

9 Bayes-Stein short sale constraint BSsc (N2 + 3N)/2

10 Mean-variance long-short sale constraint MVlsc (N2 + 3N)/2Optimal Combinations of Strategies

11 Mean-variance and minimum-variance MV/MinV (N2 + 3N)/2

12 Equally-weighted and minimum-variance EW/MinV (N2 + N)/2

13 Equally-weighted and mean-variance EW/MV (N2 + 3N)/2

14 Equally-weighted and mean-variance andminimum-variance

EW/MV/MinV (N2 + 3N)/2

Timing Strategies15 Volatility timing VT N16 Reward-to-risk timing RRT 2N

Second, prior research is updated by also considering rather newly proposed strategiessuch as the volatility and reward-to-risk timing. Furthermore, different theoretically ap-pealing models such as the minimum-variance strategy are included to serve as equalcompetitors in the horse race between optimized portfolio strategies. Above all, the naivediversification approach is empirically very successful in terms of out-of-sample portfolioperformance although completely lacking any inherent optimization and thus defines thebenchmark to beat. Table 4.1 presents an overview of the optimized portfolio strategiesconsidered in this analysis. All models are assigned a class that describes the underlyingapplied technique of creating the specific portfolios. However, this classification is notmutually exclusive such that models might also be attributed to different classes. Still,the intention is to provide a first impression on the different characteristics inherent in themodels under consideration. In addition, Table 4.1 also displays the number of estimated


parameters needed to implement a specific model under the assumption of an investmentuniverse consisting of N different risky assets. These numbers are very homogenousthroughout the strategies and mainly adopt the value of N2+3N

2 which corresponds to thenumber of estimates needed to establish the mean-variance strategy or N2+N

2 which refersto the number of estimates needed for the implementation of the minimum-variance strat-egy. This information, however, is only of quantitative nature and completely lacks qual-itative aspects. To be precise, the difference in the number of parameters needed to beestimated between the mean-variance and the minimum-variance approach is rather lim-ited and only amounts to N. Nevertheless, this difference represents the N arguably mostdifficult expected return estimations such that the gain in estimation accuracy by exclud-ing these N return estimates might be larger than ex ante expected by only looking at thequantitative differences. In addition, some models repeatedly utilize the estimated param-eters and hence potential estimation errors can have more adverse effects in comparisonto only a limited or even single usage of estimation results. The following presentationof the optimized portfolio strategies predominantly focuses on the respective underlyingidea of optimization and states the parameters needed to implement a specific strategy.Explicit optimization procedures are note stated but can be reviewed by consulting thecorresponding references mentioned in the text.

4.4.1. Equally-Weighted

The equally-weighted (EW) approach or simply naive diversification or 1/N describes aschedule that invests an equal share of 1/N in each of the N available risky assets, i.e.xi = 1/N, i = 1, . . . ,N. The relative importance of this investment approach lies in itssimplicity and widespread use. By studying the American 401(k) defined contributionsaving plan which is characterized by the fact that participants themselves decide on theirretirement saving schedule and thus pension income, Benartzi and Thaler (2001) findthat a considerable fraction of participants equally distribute their contributions across theavailable investment opportunities. This result shows evidence that investment decisionseven as important as deciding on the financial security for retirement do not necessarilyrely on more sophisticated portfolio strategies. Huberman and Jiang (2006) subsequentlyadd some more information and show that participants hardly ever invest in more than fourfunds irrespective of the actual number of available investment opportunities. Benartziand Thaler (2007), in turn, discover that the part of the sign-up form used to indicatein which funds to invest only consists of four lines. Hence, to choose more funds anadditional form needs to be requested representing an effort that might hinder furtherinvestments. This observation can explain the limited number of chosen funds, however,it does not rationalize the widespread use of equally dividing the investment. In particular,the question arises whether people even have a sufficient degree of financial literacy andwillpower to consistently execute more sophisticated strategies. And yet, nothing hasbeen said about the actual performance of the naive diversification approach relative to


optimized portfolio strategies. To start with, some thoughts about diversification.Diversification in general can result in a reduction of a portfolio’s overall volatility

hence representing a desired effect. The variance of a portfolio σ2p(x1, . . . , xN) consisting

of N risky assets can be stated as

σ2p =

N∑i=1

x2iσ

2i +

N∑i=1

N∑j=1j,i

xix jσiσ jρi j (4.9)

and thus is not only the sum of the portfolio’s individual assets but also depends on thecovariance. In fact, the correlation coefficient ρ is the key component that rationalizesthe possibility of diversification in the first place. Without loss of generality, assumethat xi > 0 and x j > 0, i, j = 1, . . . ,N, and 0 ≤ ρi j ≤ 1 for all i and j, i , j. Ingeneral, diversification benefits increase the lower the correlation between the assets sothat the minimum variance is obtained for ρi j = 0. However, also the number of assetsused to create a specific portfolio heavily influences the level of overall risk. In case ofnaive diversification, i.e. xi = 1/N, and under the assumptions of homogenous individualvariances, i.e. σ2

i = σ2 for all i, and homogenous correlation coefficients, i.e. ρi j = ρ forall i and j, i , j, Eq. (4.9) reduces to

σ2p = N

1N2σ

2 + (N2 − N)1

N2σ2ρ

=1Nσ2 + σ2ρ −

1Nσ2ρ. (4.10)

The first derivative of Eq. (4.10) with respect to the number of assets N negatively dependson the expression 1/N2, i.e. including more assets to the portfolio keeps reducing the risk,but this effect decreases with an increasing number of assets. Utilizing Eq. (4.10), it isalso true that

limN→∞

σ2p = σ2ρ = Cov(ri, r j). (4.11)

Fig. 4.1a depicts this relationship by plotting the standard deviation σp of a naive diversi-fied portfolio against the number of assets included in the portfolio. Based on Eq. (4.10),it can be seen that in case of perfect positive correlation, i.e. ρ = 1, the variance of theportfolio is equal to the assumed homogenous variance of σ = 0.2 and no diversificationeffect can be seen. In case of no correlation at all, i.e. ρ = 0, the variance vanishes inthe limit of N → ∞ and no risk remains. The intermediate case considers a correlationcoefficient of ρ = 0.64. According to Eq. (4.11) and for N → ∞, the portfolio’s standarddeviation σp is at most 1 −

√ρ percentage smaller relative to σ and hence Eq. (4.11)

establishes an upper bound for the diversification effect. The maximum reduction of theportfolio’s standard deviation in this scenario therefore amounts to 20% of σ = 0.2. In-deed, Fig. 4.1a illustrates the stagnation of risk reduction at a standard deviation level of0.16 as the number of assets increases. This level defines the cutoff between the unsystem-atic or idiosyncratic risk that can be diversified away by increasing the number of assets


0 5 10 15 20 25 30 35 40 45 500

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

Number of Assets N

Port

folio

Stan

dard

Dev

iatio

nσ

p

Effects of Naive Diversification, Number of Assets

ρ = 1ρ = 0.64ρ = 0

(a) Effects of naive diversification with focus on thenumber of assets. The graph contains three examplesfor different correlation parameter values of ρ = 1,ρ = 0.64, and ρ = 0, respectively.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

Correlation Coefficient ρ

Port

folio

Stan

dard

Dev

iatio

nσ

p

Effects of Naive Diversification, Correlation Coefficient

N = 3N = 30N = 300f (ρ) = 0.2

√ρ

(b) Effects of naive diversification with focus on thecorrelation coefficient. The graph contains three ex-amples for different numbers of assets with N = 3,N = 30, and N = 300, respectively. In addition, thefunction f (ρ) = 0.2

√ρ serves as a reference for the

case of N → ∞.Figure 4.1: Effects of Naive Diversification on the Portfolio’s Standard Deviation. This figure showseffects of naive diversification with respect to the portfolio’s standard deviation. It is assumed that all Nassets have identical variances, i.e. σ2

i = σ2 for all i, and identical correlation coefficients, i.e. ρi j = ρ forall i and j, i , j, with i, j = 1, . . . ,N. The standard deviation is set to σ = 0.2.

included in the portfolio and the systematic or market risk of 0.16 that cannot be diver-sified away. Fig. 4.1a also suggests that a sound diversification can already be achievedby investing in approximately 20 assets. However, this picture is a bit oversimplified inthe sense that empirically more assets are required to obtain well-diversified portfolios.Respecting transaction costs, Statman (1987) states that at least 30 stocks are needed toestablish a sufficient diversified portfolio. This number is obtained by comparing the costsof diversification, i.e. transactions costs, to the benefits of diversification, i.e. reductionin risk. Adding more assets to the portfolio is beneficial as long as the benefits of doingso outweigh the costs. A more general proof that diversification pays out can be found inSamuelson (1967).

Fig. 4.1b depicts the relation between the portfolio’s standard deviation and the correla-tion coefficient ρ. As the correlation coefficient approaches zero, the portfolio’s standarddeviation is decreasing. This observation is independent of the number of assets. Basedon Eq. (4.11), the standard deviation of the portfolio σp converges to σ

√ρ for N → ∞.

Since the square root is a concave function, the diversification effect and thus the decreasein risk increases relatively more for low values of ρ.

To sum up, the naive diversification approach is not only effectively used in real-worldinvestment decisions but also theoretically reasonable at least in terms of diversificationaspects. The naive diversification approach is a portfolio strategy that, at any date t,invests 1/N in each of the N available risky assets, i.e. xEW

t = 1/N ∈ RN×1. As aresult, this strategy is completely independent of the underlying data and does not requireany estimation procedure thus lacking any sort of estimation risk. The relative vector ofportfolio weights is given by


ωEWt =

xEWt∣∣∣1T

N xEWt

∣∣∣ =

1N 1N∣∣∣1TN

1N 1N

∣∣∣ =1N

1N . (4.12)

4.4.2. Sample-Based Mean-Variance

The weights of the sample-based mean-variance (MV) strategy are in accordance withEq. (4.8), i.e. the relative vector of portfolio weights is

ωMVt =

Σ−1t µt∣∣∣1T

NΣ−1t µt

∣∣∣ . (4.13)

In fact, based on an estimation period of M observations, the excess mean return and thevariance-covariance matrix are substituted with their corresponding sample counterpartssuch that the excess sample mean return µt is calculated as

µt =1M

t∑τ=t−M+1

µτ (4.14)

and the sample variance-covariance matrix Σt is obtained via

Σt =1

M − N − 2

t∑τ=t−M+1

(µτ − µt) (µτ − µt)T . (4.15)

The particular form of the sample variance-covariance matrix is chosen based on the factthat its inverse Σ−1

t is an unbiased estimator of Σ−1t in case of M > N + 4, see Kan and

Zhou (2007). This strategy does not incorporate any specific estimation technique andtherefore is exposed to potential estimation errors.

4.4.3. Minimum-Variance

The minimum-variance (MV) strategy refers to the optimization problem given by

minωMinV

t

12

(ωMinVt )T Σtω

MinVt subject to 1T

NωMinVt = 1. (4.16)

The solution to this optimization problem is

ωMinVt =

Σ−1t 1N

1TNΣ−1

t 1N, (4.17)

see Appendix C. This strategy is implemented by substituting the variance-covariance ma-trix with its sample-based counterpart Σt. In contrast to the sample-based mean-variancestrategy, the weights of the minimum-variance strategy are only dependent on the samplevariance-covariance matrix and avoid estimating the arguably most difficult and criticalexpected returns. Therefore, this strategy is relatively less exposed to estimation errors.


In fact, the minimum-variance portfolio strategy does not generally belong to the classof optimized approaches with respect to the mean-variance trade-off. This would be onlytrue in case of identical expected returns across all assets. Nevertheless, the minimum-variance strategy is worth to be considered due to its theoretically limited estimation errorand because some of the following optimized portfolio strategies do incorporate this strat-egy.

4.4.4. Value-Weighted

The two-fund separation theorem according to Tobin (1958) states that there exists aunique efficient risky market portfolio which every mean-variance investor should hold.The separation of the total investment between this risky market portfolio and the risk-free asset is then only dependent on the investor’s individual risk-aversion parameter γ.The CAPM by Sharpe (1964), Lintner (1965), and Mossin (1966) can be understood asa model whose equilibrium outcome coincides with this very efficient risky market port-folio. In fact, it is exactly the value-weighted market portfolio in which each asset isrepresented according to its relative value of market equity. The expected excess returnof this value-weighted (VW) portfolio strategy is directly obtained from the data sets andgiven by the US and European MKT series, respectively.

4.4.5. MacKinlay-Pástor

MacKinlay and Pástor (2000) start from the assumption that asset returns can be perfectlydescribed by factor models such as described in Section 4.3. However, in most or even allreasonably scenarios the exact model is not known, i.e. risk factors are missing. Never-theless, the mispricing of asset returns based on the lack of a risk factor translates to thevariance-covariance matrix of the residuals and this knowledge can be exploited to im-prove the estimation of the expected returns. By doing so, the variance-covariance matrixof the returns transforms more and more into a simple identity matrix of the form σ2

t IN thehigher the level of uncertainty about the underlying asset pricing factor model. Accord-ing to the MacKinlay and Pástor (2000) (MP) strategy, the variance-covariance matrix ofreturns is expressed as

ΣMPt = αtµtµ

Tt + σ2

t IN (4.18)

where αt is a term that represents the model misspecification. The optimal weights canthen be reformulated as

xMPt =

1γ

(ΣMPt )−1µMP

t

=1γ

1σ2

t

(IN −

αtµMPt (µMP

t )T

σ2t + αt(µMP

t )TµMPt

)µMP

t

=1γ

1σ2

t + αt(µMPt )TµMP

tµMP

t . (4.19)


The estimates for αt, µMPt , and σ2

t can be obtained via the process of a likelihood max-imization, however, this demanding optimization can be avoided by applying a semi-analytical solution as presented in Tu and Zhou (2011) yielding

xMPt ≈

1γ

µt

λ1,t − µTt µt

. (4.20)

In this context, λ1,t denotes the largest eigenvalue and v1,t the corresponding eigenvectorof the matrix ΣML

t + µMLt (µML

t )T , where the maximum likelihood (ML) estimates of thesample mean and the variance-covariance matrix are given by µML

t and ΣMLt , respectively.

The adjusted return is expressed as µt = v1,tµMLt v1,t such that the relative vector of portfolio

weights can be stated as

ωMPt =

xMPt∣∣∣1T

N xMPt

∣∣∣ =

1γ

µt

λ1,t−µTt µt∣∣∣∣1T

N1γ

µt

λ1,t−µTt µt

∣∣∣∣ =µt∣∣∣1TN µt

∣∣∣ . (4.21)

4.4.6. Bayes-Stein

The Bayes-Stein (BS) portfolio strategy is a combination of shrinkage and Bayesian esti-mation. Following Jorion (1986), the expected sample mean return estimator is obtainedby shrinking the expected sample mean return toward the mean of the expected samplereturn estimate of the minimum-variance portfolio µMinV

p,t = µTt ω

MinVt , i.e.

µBSt = (1 − φt)µt + φtµ

MinVp,t , (4.22)

where the shrinkage factor 0 < φt < 1 takes the form

φt =N + 2

(N + 2) + M(µt − µMinVp,t 1N)T Σ−1

t (µt − µMinVp,t 1N)

. (4.23)

Supported by empirical evidence, Jorion (1985) argues that shrinking the expected samplemean toward the minimum-variance portfolio improves the estimation due to the fact thatthe minimum-variance portfolio is the optimal choice in case of identical expected returnsof all assets. The Bayes-Stein approach additionally estimates the variance-covariancematrix by applying a traditional Bayesian estimation technique to account for potentialestimation error. This procedure leads to an estimate of the form

ΣBSt = Σt

(1 +

1M + κt

)+

κt

M(M + 1 + κt)1N1T

N

1TNΣ−1

t 1N(4.24)

with κt = Mφt/(1 − φt) being the precision parameter of the prior

p(µt

∣∣∣Σt, µMinVp,t , κt) ∝ exp

(−

12

(µt − µMinVp,t 1N)T (κtΣ

−1t )(µt − µ

MinVp,t 1N)

). (4.25)


As a result, the solution of the underlying optimization problem stated in Eq. (4.7) trans-forms to

xBSt =

1γ

(ΣBSt )−1µBS

t . (4.26)

The relative vector of portfolio weights is given by

ωBSt =

xBSt∣∣∣1T

N xBSt

∣∣∣ =

1γ(ΣBS

t )−1µBSt∣∣∣∣1T

N1γ(ΣBS

t )−1µBSt

∣∣∣∣ =(ΣBS

t )−1µBSt∣∣∣1T

N(ΣBSt )−1µBS

t

∣∣∣ . (4.27)

4.4.7. Portfolio Weight Constraints

The portfolio weight constraints considered in this analysis result in four specifications.On the one hand, short sale constraints are applied to the sample-based mean variance(MVsc), minimum-variance (MinVsc), and Bayes-Stein (BSsc) strategy, respectively, i.e.all weights are forced to adopt positive values. As a result, the mean-variance optimizationproblem alters to a quadratic optimization problem including linear inequality constraintsand can be stated as

maxxt

E(u(xt)

)= xT

t µt −γ

2xT

t Σtxt subject to xt ≥ 0. (4.28)

Unfortunately, there exists no closed-form solution to this optimization problem such thata standard numerical quadratic optimization program is applied to obtain the respectivefinal weight vectors.

DeMiguel et al. (2009b) show that adding a nonnegativity weight constraint to themean-variance strategy is effectively a form of shrinkage of the expected returns towardthe average of all the expected returns. In the same spirit, Jagannathan and Ma (2003) ar-gue that the nonnegativity weight restriction of the minimum-variance strategy in factshrinks the larger elements of the variance-covariance matrix toward zero and henceresults in similar outcomes as the approaches focusing solely on the shrinkage of thevariance-covariance matrix such as in Ledoit and Wolf (2004). As always, shrinking canpotentially affect the result in two ways. Specification error introduced by the shrinkingprocess can render the results biased, whereas the level of estimation error decreases rel-ative to nonshrinking. This trade-off is assumed to be in favor of the shrinking techniquesuch that the gain in estimation accuracy outweighs the loss incurred from obtaining bi-ased results.

In addition, a long-short sale constraint is considered that not only constraints shortsales but also long sales. This strategy is often applied in the context of mutual and hedgefunds and arguably represents are more realistic scenario relative to no constraints at allor short sale only constraints, see also Michaud (1993) and Sorensen et al. (2007). In fact,these portfolio weight constraint strategies might come very close to the truth in termsof mimicking actual investment behavior. The specific long-short sale constraint strategy


considered in this analysis takes the form

maxxt

E(u(xt)

)= xT

t µt −γ

2xT

t Σtxt subject to − 0.3 ≤ xt ≤ 1.3. (4.29)

This represents the so-called 130/30 strategy, i.e. the portfolio can be exposed up to130% in long equity positions and up to 30% in short equity positions such that the totalexposure to equity still amounts to a 100% long position. This strategy is only consideredwith respect to the sample-based mean-variance strategy and is referred to as MVlsc.

4.4.8. Mean-Variance and Minimum-Variance

Kan and Zhou (2007) argue that in case of estimation uncertainty with respect to thedata generating parameters, it is not optimal to hold the efficient market portfolio of riskyassets in combination with the risk-free asset as proposed by Markowitz (1952) and To-bin (1958). The idea is that when estimating the efficient portfolio with error, includingadditional portfolios could be beneficial in terms of reducing estimation risk through di-versification as long as the estimation errors of the efficient market portfolio and the newlyintroduced portfolio are not perfectly correlated. In particular, including a third portfolioto the risky and risk-free assets, the minimum-variance portfolio, can help to reduce es-timation risk. The minimum-variance portfolio is chosen because it does not rely on theestimation of the expected mean returns and hence can be estimated with a relatively highprecision. Kan and Zhou (2007) refer to this strategy as three-fund separation and for-mulate the weights as a weighted sum of the sample-based mean-variance portfolio thatalready incorporates the risk-free asset and the minimum-variance portfolio (MV/MinV),i.e.

xMV/MinVt =

1γ

(ctΣ−1t µt + dtΣ

−1t 1N

). (4.30)

The individual weights ct and dt are chosen so as to maximize the expected out-of-sampleperformance with respect to the utility of a mean-variance investor. It is

ct =(M − N − 1)(M − N − 4)

M(M − 2)

ψ2adj.,t

ψ2adj.,t + N

M

(4.31)

and

dt =(M − N − 1)(M − N − 4)

M(M − 2)

NM

ψ2adj.,t + N

M

µMinVp,t (4.32)

with µMinVp,t = µT

t ωMinVt and

ψ2adj.,t =

(M − N − 1)ψ2t − (N − 1)

M+

2(ψ2t )

N−12 (1 + ψ2

t )−M−2

2

MBψ2t /(1+ψ2

t )

(N−1

2 , M−N+12

) (4.33)


being an adjusted estimator of

ψ2t = (µt − µ

MinVp,t 1N)T Σ−1

t (µt − µMinVp,t 1N). (4.34)

The term Bψ2t /(1+ψ2

t )

(N−1

2 , M−N+12

)refers to the incomplete beta function

Bx(a, b) =

x∫0

ya−1(1 − y)b−1dy. (4.35)

The first fraction of the adjusted estimator ψ2adj.,t represents the unbiased estimator of ψ2

t

and the second fraction an adjustment in case the unbiased estimator is too small. Theresulting portfolio can thus be characterized by

xMV/MinVt =

1γ

(M − N − 1)(M − N − 4)M(M − 2)

ψ2adj.,t

ψ2adj.,t + N

M

Σ−1t µt +

NM

ψ2adj.,t + N

M

µMinVp,t Σ−1

t 1N

.(4.36)

The relative vector of portfolio weights takes the form

ωMV/MinVt =

xMV/MinVt∣∣∣1T

N xMV/MinVt

∣∣∣=

1γ

(M−N−1)(M−N−4)M(M−2)

((ψ2

adj.,t

ψ2adj.,t+

NM

)Σ−1

t µt +

(NM

ψ2adj.,t+

NM

)µMinV

p,t Σ−1t 1N

)∣∣∣∣∣1T

N1γ

(M−N−1)(M−N−4)M(M−2)

((ψ2

adj.,t

ψ2adj.,t+

NM

)Σ−1

t µt +

(NM

ψ2adj.,t+

NM

)µMinV

p,t Σ−1t 1N

)∣∣∣∣∣=

(ψ2

adj.,t

ψ2adj.,t+

NM

)Σ−1

t µt +

(NM

ψ2adj.,t+

NM

)µMinV

p,t Σ−1t 1N∣∣∣∣∣1T

N

((ψ2

adj.,t

ψ2adj.,t+

NM

)Σ−1

t µt +

(NM

ψ2adj.,t+

NM

)µMinV

p,t Σ−1t 1N

)∣∣∣∣∣ (4.37)

Intuitively, the weight put on the minimum-variance portfolio is an increasing functionin the number of assets and a decreasing function in the number of observations as bothscenarios tend to increase the estimation error when calculating the efficient portfolio. Ahigher risk of estimation error leads to a greater reliance on the minimum-variance port-folio that can be calculated with a relatively higher accuracy. However, it is important tonote that the determination of the weights ct and dt regarding the optimal combination ofthe optimal portfolios is also exposed to estimation risk. In fact, as long as the processof optimally combining already optimal strategies introduces an additional level of esti-mation risk, it is not ex ante clear that these potential errors are completely offset by thetheoretically predicted diversification gains.

4.4.9. Equally-Weighted and Minimum-Variance

This strategy as proposed in DeMiguel et al. (2009b) follows the idea of Kan and Zhou(2007). However, instead of optimally combining the mean-variance portfolio strategy


with the minimum-variance equivalent, DeMiguel et al. (2009b) focus on the optimalcombination of the naive diversification approach and the minimum-variance strategy(EW/MinV). Since the former does not rely on any estimation procedure at all and theminimum-variance strategy only deals with uncertainty regarding the estimation of thevariance-covariance matrix, the strategy as suggested by DeMiguel et al. (2009b) is sup-posed to be exposed to only very limited estimation risk. The relative vector of portfolioweights is given by

ωEW/MinVt = ct

1N

1N + dtΣ−1t 1N (4.38)

subject to 1TNω

EW/MinVt = 1. The individual weights ct and dt are chosen so as to maximize

the expected out-of-sample performance with respect to the utility of a mean-varianceinvestor. It is

ct = 1 − dt1TNΣ−1

t 1N (4.39)

and

dt =(M − N − 2)(1T

NΣt1N)(1TNΣ−1

t 1N) − N2M

N2(M − N − 2)k(1TNΣ−1

t 1N) − 2MN2(1TNΣ−1

t 1N) + (M − N − 2)(1TNΣ−1

t 1N)2(1TNΣt1N)

(4.40)with k =

M2(M−2)(M−N−1)(M−N−2)(M−N−4) .

4.4.10. Equally-Weighted and Mean-Variance

In the spirit of Kan and Zhou (2007) and DeMiguel et al. (2009b), Tu and Zhou (2011)propose a new set of combinations of optimal portfolio strategies. First, the naive diver-sification approach is combined with the mean-variance strategy (EW/MV). The optimalcombination is determined by the coefficient δ that is chosen so as to maximize the ex-pected out-of-sample performance with respect to the utility of a mean-variance investor,i.e.

xEW/MVt = (1 − δt)ωEW

t + δωMVt . (4.41)

The combination of the mean-variance strategy with the equally-weighted approach canbe understood as a shrinkage toward naive diversification resulting in a trade-off betweenbias and estimation precision. In this context, the shrinkage toward the naive diversifica-tion approach will bias the plain mean-variance solution but render the estimation moreprecise. Since the optimal combination parameter δ is also subject to estimation risk, itis ex ante not clear, that the resulting out-of-sample excess return performance will beindeed superior to either the naive diversification approach or the mean-variance strategy.Tu and Zhou (2011), however, attribute a rather limited estimation risk to the determina-tion of δ such that a superior performance of their strategy is expected. It is

δt =π1,t

π1,t + π2,t(4.42)


withπ1,t = (ωEW

t )T ΣtωEWt −

2γ

(ωEWt )T µt +

1γ2 θ

2adj.,t (4.43)

andπ2,t =

1γ2 (c1 − 1)θ2

adj.,t +c1

γ2

NM, (4.44)

with c1 =(M−2)(M−N−2)

(M−N−1)(M−N−4) , where θ2adj.,t is an adjusted estimator of θ2

t = µTt Σ−1

t µt satisfying

θ2adj.,t =

(M − N − 2)θ2t − N

M+

2(θ2t )

N2 (1 + θ2

t )−M−2

2

MBθ2t /(1+θ2

t )

(N2 ,

M−N2

) . (4.45)

The incomplete beta function Bθ2t /(1+θ2

t )

(N2 ,

M−N2

)is defined as in Eq. (4.35). The condition

T > N +4 needs to be fulfilled to ensure the existence of the second moment of the inverseof the variance-covariance matrix. The relative vector of portfolio weights is given by

ωEW/MVt =

xEW/MVt∣∣∣1T

N xEW/MVt

∣∣∣ =(1 − δt)ωEW

t + δωMVt∣∣∣∣1T

N

((1 − δt)ωEW

t + δωMVt

)∣∣∣∣ . (4.46)

4.4.11. Equally-Weighted and Mean-Variance and Minimum-Variance

A second combination of optimal portfolio strategies proposed by Tu and Zhou (2011)is the extension of the already optimally combined MV/MinV strategy by Kan and Zhou(2007) to the inclusion of the naive diversification approach (EW/MV/MinV). To be pre-cise, it is

xEW/MV/MinVt = (1 − δt)ωEW

t + δtωMV/MinVt , (4.47)

where δt is again chosen so as to maximize the expected out-of-sample performance withrespect to the utility of a mean-variance investor and takes the form

δt =π1,t − π13,t

π1,t − 2π13,t + π3,t. (4.48)

It is π1,t as in Eq. (4.43) and

π13,t =1γ2 θ

2adj.,t −

1γ

(ωEWt )T µt

+1γc1

( (ψ2

adj.,t(ωEWt )T µt + (1 − ψ2

adj.,t)µMinVp,t (ωEW

t )T 1N

)−

1γ

(ψ2

adj.,tµTt Σ−1

t µt + (1 − ψ2adj.,t)µ

MinVp,t µT

t Σ−1t 1N

) )(4.49)

andπ3,t =

1γ2 θ

2adj.,t −

1γ2c1

(θ2

adj.,t −NMψ2

adj.,t

)(4.50)


with ψadj.,t as in Eq. (4.33), θadj.,t as in Eq. (4.45), and c1 =(M−2)(M−N−2)

(M−N−1)(M−N−4) . The relativevector of portfolio weights is given by

ωEW/MV/MinVt =

xEW/MV/MinVt∣∣∣1T

N xEW/MV/MinVt

∣∣∣ =(1 − δt)ωEW

t + δωMV/MinVt∣∣∣∣1T

N

((1 − δt)ωEW

t + δωMV/MinVt

)∣∣∣∣ . (4.51)

4.4.12. Volatility Timing

The volatility timing (VT) strategy by Kirby and Ostdiek (2012) can be stated as

ωVTit =

(1/σ2it)η

N∑i=1

(1/σ2it)η

with i = 1, . . . ,N, (4.52)

where σit is the sample volatility of the ith excess risky return and η ≥ 0. This ap-proach coincides with the solution of the mean-variance strategy when ignoring expectedreturns and under the assumption of a diagonal variance-covariance matrix. Assuminga diagonal variance-covariance matrix, however, is equivalent to assuming all the pair-wise correlation coefficients of risky asset to be zero and thus representing an extremeform of shrinking the variance-covariance matrix to a diagonal. The potential bias thatcomes with this form of extreme shrinkage is opposed to the estimation precision im-provement as there are N(N − 1)/2 parameters less to estimate. The assumption, as usual,is that gains in estimation precision outweigh the incurred bias. The strategy of Kirbyand Ostdiek (2012) comes with some more advantages. In fact, the weights proposedin Eq. (4.52) neither result in negative weights nor do they rely on the inversion of thevariance-covariance matrix, two characteristics that potentially lead to a higher estima-tion accuracy. The parameter η can be understood as a tuning parameter that decideson the sensitivity of the portfolio weights with respect to changes in volatility. In thelimit of η → 0, the portfolio weights coincide with the weights of the naive diversifica-tion approach. The benchmark case representing the mean-variance approach under theassumption of a diagonal variance-covariance matrix is η = 1.

4.4.13. Reward-to-Risk Timing

This strategy is also proposed by Kirby and Ostdiek (2012) and related to their VT strat-egy. However, the reward-to-risk timing (RRT) strategy also accounts for expected re-turns, i.e.

ωRRTit =

(max(µit, 0)/σ2

it)η

N∑i=1

(max(µit, 0)/σ2

it)η with i = 1, . . . ,N. (4.53)

In this specific context, the expected returns are restricted to be positive, i.e. the rep-resentative investor does not invest in assets that yield a negative expected return. Thisconstraint is chosen so as to ensure a high degree of comparability between the VT and

5. Methodology 29

RRT strategy. Both strategies result in positive weight vectors and hence reduce the riskof adopting extreme positions while mitigating the risk of high estimation errors at thesame time.

5. Methodology

This section is dedicated to the methodology used to perform the horse race between thedifferent portfolio strategies. In fact, the data sets employed in this analysis are dividedinto an in-sample period used for estimating the data generating parameters and an out-of-sample period for which monthly excess returns are determined. Subsequently, theseout-of-sample return forecasts are evaluated based on different performance evaluationcriteria. To start with, the following describes the actual estimation procedure based on arolling-window approach. Next, the employed performance evaluation criteria are intro-duced and discussed.

5.1. Estimation Procedure

The estimation procedure of the out-of-sample forecasting is based on a rolling windowapproach and can be described as a four-step procedure. First, the length of the estima-tion window M needs to be defined. Here, the estimation window consists of M = 120consecutive monthly excess return observations corresponding to 10 years of return data.Second, starting at time t = M, all parameters required for the calculation of the optimalportfolio weights ωi

t at time t for strategy i, i = EW,MV, . . . ,RRT, are estimated based onthe M previous months, i.e. [t − M + 1, . . . , t]. Typically, these parameters are the samplemean and the variance-covariance matrix. Third, the optimal portfolio weights ωi

t are ob-tained and used to calculate the corresponding estimated out-of-sample portfolio returnsfor time t + 1. Fourth, the estimation window is shifted ahead one period by including thenext return observation corresponding to date t + 1 and dropping the earliest return so asto keep the length of the estimation window constant, i.e. [t − M + 2, . . . , t + 1]. Finally,the process continues with step two and rolls through the data until the end of the returnseries is reached. Given a time series of size T , this approach results in an estimatedout-of-sample return series of length T − M.

The rolling-window approach has several degrees of freedom that can potentially affectthe final results. It is ex ante not evident, for example, how to optimally choose thelength of the estimation window. A shorter estimation window might lead to inaccurateestimations of the data generating parameters, whereas a longer estimation window mightbe insensitive to potential structural breaks, i.e. time variations in the sample moments.The main motivation for applying the rolling window approach of size M = 120 is givenby the fact that this appears to be the working horse of the related empirical literature andhence preserves comparability. Shorter and longer estimation windows are included inthe robustness checks.

5. Methodology 30

5.2. Performance Evaluation Criteria

Several different performance evaluation criteria are employed in this analysis to assessthe actual performance of the out-of-sample monthly excess return series obtained fromthe rolling window approach with respect to the optimized portfolio strategies introducedin Section 4.4. The majority of performance criteria fall into the classification of reward-to-risk ratios of which the Sharpe ratio is arguably the most famous and widely used.However, out-of-sample return forecasts typically result in time series that are not nor-mally distributed. For this reason, this analysis also considers reward-to-risk ratios thatdo not critically rely on this normality assumption of asset returns. In fact, whereas the re-ward measures throughout all ratios exclusively focus on the out-of-sample mean returns,the risk measures incorporate several competing approaches and hence create differentrisk-adjusted performance measures. Moreover, the focus is set on reward-to-risk ratiosthat do not excessively rely on the estimation of additional quantities which might fur-ther question the reliability of the obtained results such as for example the Treynor ratio.Following Frahm et al. (2012), this analysis does also not consider turnover which em-pirically is a common performance measure providing information about the transactioncosts of realizing the respective portfolios in every period. The reason for not includingthis measurement lies in the fact that the setting of this analysis is a rather static one fo-cusing on a repeated single period portfolio choice problem and thus does not actuallyincorporate the very nature of portfolio rebalancing. Instead, the certainty equivalent re-turn is included which provides information about the risk-free return a representativeinvestor demands to be indifferent between this risk-free return and the correspondingreturns resulting from the respective different portfolio strategies.

5.2.1. Sharpe Ratio

Introduced in Sharpe (1966) and generalized in Sharpe (1994), the out-of-sample Sharperatio (SR) of portfolio strategy i, i = EW,MV, . . . ,RRT, is defined as

SRi=

E(Rip) − r f

σip

=µi

p

σip. (5.1)

The numerator yields the difference between the expected out-of-sample nonexcess meanportfolio returns for strategy i given by E(Ri

p) and the risk-free rate r f , i.e. µip = E(Ri

p)−r f ,or rather

µip =

1T − M

T∑t=M+1

µit, (5.2)

where µip denotes the expected out-of-sample excess mean portfolio return. The denomi-

nator contains the corresponding sample portfolio standard deviation

5. Methodology 31

σip =

√√1

T − M − 1

T∑t=M+1

(µit − µ

ip)2. (5.3)

A higher positive Sharpe ratio is preferred over a lower positive Sharpe ratio as the for-mer yields a more favorable return-to-risk trade-off. Negative Sharpe ratios indicate anoutperformance of the portfolio strategy by the risk-free asset and thus represent an un-desired event. As a further, however, not attainable benchmark, the in-sample (IS) Sharperatio based on the in-sample mean and variance-covariance matrix, i.e. µIS and ΣIS, iscalculated as

SRMVIS =

µMVp,IS

σMVp,IS

=µT

ISωMVIS√

(ωMVIS )T ΣISω

MVIS

. (5.4)

The in-sample weights are denoted by ωMVIS based on a time period of [M + 1, . . . ,T ],

i.e. the originally out-of-sample period is now treated as if it were known in advance andhence optimization on this period results in the highest possible Sharpe ratio. In the samemanner, the remaining performance evaluation criteria are evaluated with respect to theirin-sample performance.

The Sharpe ratio is also not free from criticism. First, it assumes normally distributedreturns. Hence, time series of returns that violate this assumption might lead to biasedestimates of Sharpe ratios. Second, risk is represented by the standard deviation, i.e.volatility. Volatility itself, however, is not necessarily an unwanted characteristic butrather a positive feature when restricted on its upside potential. The price function ofa long call option, for example, positively depends on the volatility of the underlyingmarket and illustrates the beneficial properties that volatility can have. Still, its concep-tual simplicity and widespread use render the Sharpe ratio a valuable tool for evaluatingrisk-adjusted portfolio performance.

To finally answer the question of whether there are optimized portfolio strategies thatcan consistently outperform the naive diversification approach, it is necessary to estab-lish a statistically informed decision criterion. As a first step toward this direction andin accordance with DeMiguel et al. (2009b), the test statistic proposed by Jobson andKorkie (1981) (JK) adjusted for the correction suggested by Memmel (2003) is appliedto decide on the pairwise difference between the out-of-sample Sharpe ratio of the naivediversification approach, SREW, and strategy i, SRi, with i = MV,MinV, . . . ,RRT. It is

SREW− SR

i∼ N(0, Θ) (5.5)

with

Θ =1

T − M

(2(1 − (ρEW,i)2

)+

12

((SR

EW)2 + (SR

i)2 − 2SR

EWSR

i(ρEW,i)2

)). (5.6)

The out-of-sample correlation coefficient of the two portfolios under consideration is de-

5. Methodology 32

noted by ρEW,i. The null hypotheses of the one-sided tests of the form Hi0 : SR

EW−SR

i≥ 0

are then decided on the test statistics

ziJK =

SREW− SR

i√Θ

. (5.7)

These test statistics refer to a sample size of T −M. Given the assumption of independentand identically normally distributed out-of-sample portfolio returns, the test statistics zi

JK

are asymptotically distributed as a standard normal. However, out-of-sample portfolioreturns are at best approximately normally distributed but in general rather display fattails and resemble a leptokurtic distribution. This, in turn, might lead to wrong inferenceswith respect to p-values. To account for this drawback, this analysis also considers astudentized time series bootstrap approach based on the Delta method and as introducedby Ledoit and Wolf (2008)4 (LW) that is independent of the normality assumption. Let

f (ν) = SREW− SRi (5.8)

denote the difference between the Sharpe ratios of the naive diversification approach andthe optimized strategy i with ν = (µEW, µi, γEW, γi) and the raw variances γEW = E

((rEW)2)

and γi = E((ri)2). It follows that

f (µEW, µi, γEW, γi) =µEW√

γEW − (µEW)2−

µi√γi − (µi)2

. (5.9)

According to Ledoit and Wolf (2008) and the Delta method, this results in

√T − M

(SR

EW− SR

i− (SREW

− SRi)) d∼ N

(0,∇T f (ν)Ψ∇ f (ν)

)(5.10)

with

∇T f (ν) =

γEW(γEW − (µEW)2)3/2 ,−

γi(γi − (µi)2)3/2 ,−

12

µEW(γEW − (µEW)2)3/2 ,

12

µi(γi − (µi)2)3/2

.(5.11)

The null hypotheses of the two-sided tests of the form Hi0 : SR

EW− SR

i= 0 are then

decided based on the test statistics

ziLW =

SREW− SR

i− (SREW

− SRi)√∇T f (ν)Ψ∇ f (ν)/(T − M)

. (5.12)

The estimate of the matrix Ψ is obtained via a heteroskedasticity and autocorrelation(HAC) robust kernel estimation and the p-values are calculated as in Ledoit and Wolf(2008) with a block length of b = 5 and K = 1, 000 bootstrap iterations.

4Code is freely available athttps://www.econ.uzh.ch/en/people/faculty/wolf/publications.html.

https://www.econ.uzh.ch/en/people/faculty/wolf/publications.html

5. Methodology 33

5.2.2. Sortino Ratio

Since individuals are typically loss-averse but not gain-averse, a downside risk measurecould be more appropriate in capturing the investor’s fundamental trade-off between re-turn and risk. Originated from Bawa (1975) and Fishburn (1977), lower partial moments(LPM) only focus on the negative deviations of returns, i.e. the downside risk, with re-spect to a minimum acceptable return (MAR) rMAR. The lower partial moment of order n

for strategy i is

LPMin(rMAR) =

1T − M

T∑t=M+1

max(rMAR − µit, 0)n. (5.13)

The order n determines the weighting of the return’s deviation from the minimum accept-able return and translates into a representation of a risk-averse investor for values largerthan one. In this analysis, the minimum acceptable return is set to zero. However, due tothe fact that µi

t refers to excess returns, it is possible to equivalently state the lower par-tial moment in terms of nonexcess returns and a minimum acceptable return equal to therisk-free rate. The Sortino ratio (SoR) is obtained by substituting the standard deviationin the denominator of the Sharpe ratio with the lower partial moment of order two. Basedon Bawa (1978), Harlow (1991), and Sortino and Van Der Meer (1991), the Sortino ratioemerges as

SoRi=

µip√

1T−M

T∑t=M+1

max(rMAR − µit, 0)2

. (5.14)

5.2.3. Omega Ratio

The Omega ratio (OR) is a measure that does not require any assumption on the underly-ing distribution of the specific time series and hence can be referred to as a nonparametricratio. In addition, the Omega ratio considers all moments of the return distribution asopposed to the Sharpe ratio that only incorporates the first two moments. Established inKeating and Shadwick (2002), the Omega ratio as formulated in Kaplan and Knowles(2004) is

ORi=

µip

1T−M

T∑t=M+1

max(rMAR − µit, 0)

+ 1. (5.15)

A higher Omega ratio is preferred over a lower Omega ratio and, for example, a returnseries yielding a higher excess kurtosis has ceteris paribus a lower Omega ratio relative toa return series with a lower excess kurtosis.

5.2.4. Calmar Ratio

The Calmar ratio (CR) is a somewhat more extreme risk-adjusted performance evaluationmeasure as it only considers the most adverse development of the out-of-sample returnseries. Nevertheless, investors might be more concerned about a single severe loss relative

5. Methodology 34

to many small losses. The Calmar ratio developed by Young (1991) is the expected excessout-of-sample mean portfolio return over the maximum drawdown (MaxDD) for strategyi, i.e.

CRi=

µip

−MaxDDip

. (5.16)

The drawdown (DD) for strategy i at time t is the percentage loss of the portfolio’s valuesince the previous peak until time t, i.e.

DDip,t = max

(0, max

τ∈[M+1,t]

(piτ − pi

t

piτ

)). (5.17)

The running maximum value of the portfolio in the time interval [M + 1, t] is denotedby pi

τ and pit is the actual value of the portfolio at time t. The maximum drawdown

for strategy i is the largest loss that can occur during the entire out-of-sample period[M + 1,T ] and hence is given by

MaxDDip = max

t∈[M+1,T ](DD

ip,t) = max

t∈[M+1,T ]

(0, max

τ∈[M+1,t]

(piτ − pi

t

piτ

)). (5.18)

Similar to the other risk-adjusted performance measures, higher values of the Calmar ratioare preferred over lower values.

5.2.5. Return on Value-at-Risk

Value-at-risk (VaR) provides information about the maximum possible loss of an invest-ment during the out-of-sample period that is not exceeded with a probability of 1 − α.Formally, the value-at-risk can be defined as

Prob(∆p ≤ −VaRp) = 1 − α, (5.19)

where p denotes the value of the portfolio. Under the assumption of normally distributedreturns, the value-at-risk can be obtained by

VaRip = −(µi

p + zασip) (5.20)

with zα representing the α-quantile of the standard normal distribution. This analysisapplies a value of α = 5%. The excess return on value-at-risk (RoVaR) as proposed byDowd (2000) is defined as

RoVaRi=

µip

VaRip

. (5.21)

The value-at-risk strategy is occasionally criticized for not being a coherent risk measureas defined in Artzner et al. (1999). In particular, the axiom of subadditivity is violated,i.e. the value-at-risk of two portfolios after they have been merged can exceed the sum

5. Methodology 35

of the individual portfolio’s values-at-risk. In addition, the value-at-risk criterion doesnot provide any information about the potential loss that might occur beyond the cut-off

threshold specified by α. The return on value-at-risk is included in this analysis due toits popular use in the financial industry with respect to risk management and regardingfinancial reporting and regulatory guidelines concerning a bank’s required capital set bythe Bank for International Settlements in their Basel Accords.

5.2.6. Certainty Equivalent Return

The certainty equivalent return (CE) is the risk-free return a representative investor de-mands to be indifferent between this risk-free return and the corresponding returns re-sulting from the different portfolio strategies, respectively. In this analysis, the certaintyequivalent is defined as in DeMiguel et al. (2009b), i.e.

CEi= µi

p −γ

2(σi

p)2. (5.22)

This actually represents the expected utility of a mean-variance investor with respect toEq. (4.6). However, this expected utility can serve as an approximation for the certaintyequivalent as shown in Christensen and Feltham (2003). In this context, the investor’srisk-aversion parameter γ is set to one.

Similar to the Sharpe ratio and also in accordance with DeMiguel et al. (2009b), theDelta method is applied to decide on the pairwise difference between the out-of-samplecertainty equivalent returns of the naive diversification approach, CEEW, and strategy i,CEi, with i = MV,MinV, . . . ,RRT. Let

f (ν) = CEEW− CEi (5.23)

with ν = (µEW, µi, γEW, γi) and the raw variances γEW = E((rEW)2) and γi = E

((ri)2).

Under the assumption of independent and identically normally distributed returns, it is

√T − M

(CE

EW− CE

i− (CEEW

− CEi)) d∼ N

(0,∇T f (ν)Ω∇ f (ν)

)(5.24)

with∇T f (ν) =

(−(1 + γµEW), 1 + γµi,

γ

2,−γ

2

)(5.25)

and

Ω =

(σEW)2 σEW,i 0 0σi,EW (σi)2 0 0

0 0 2(σEW)4 2(σEW,i)2

0 0 2(σi,EW)2 (σi)4

. (5.26)

The null hypotheses of the one-sided tests of the form Hi0 : CE

EW− CE

i≥ 0 are then


decided on the test statistics

zi∆ =

CEEW− CE

i− (CEEW

− CEi)√∇T f (ν)Ω∇ f (ν)/(T − M)

. (5.27)

These test statistics, however, also critically depend on the assumption of having normallydistributed out-of-sample excess returns. To account for this drawback, the Sharpe ratiobootstrap methodology introduced by Ledoit and Wolf (2008) is adjusted for a certaintyequivalent return setting. Hence, the null hypotheses of the two-sided tests of the formHi

0 : CEEW− CE

i= 0 are decided based on the test statistics given in Eq. (5.27) but the

matrix Ω is substituted with the HAC robust matrix Ψ. The p-values are calculated as inLedoit and Wolf (2008) with a block length of b = 5 and K = 1, 000 bootstrap iterations.

Finally, it is important to note that the underlying mean-variance framework of thisanalysis favors the Sharpe ratio as well as the certainty equivalent return relative to theother risk-adjusted performance measures. This is due to the fact that the optimized port-folio strategies are all designed so as to favor expected return and dismiss risk measuredin terms of variance of return. Since the other performance criteria measure risk not interms of variance, the portfolio strategies considered in this analysis are strictly speakingnot optimized with respect to these additional risk measures. In fact, if the representativeinvestor would only care about risk, for example, measured as the value-at-risk then theunderlying maximization problem should be altered such that the variance is substitutedwith the value-at-risk. An application of such a mean-VaR approach can be found inFabozzi et al. (2010). Nevertheless, the evaluation of optimized portfolio strategies basedon the mean-variance framework but assessed by performance evaluation criteria that notonly account for risk measured in terms of variance of return can provide valuable insightsinto the strategies’ performance behavior.

6. Empirical Results and Discussion

This analysis can be understood as a horse race between different out-of-sample optimizedportfolio strategies taking place on venues represented by different data sets where the roleof the jockeys is assigned to the different performance evaluation criteria. This sectionnow presents the results of these competitions starting with the findings regarding theSharpe ratio as displayed in Table 6.1.

A first observation refers to the consistently highest Sharpe ratios of the MV IS strategyacross all data sets indicated by the ranking -1-. This insight, however, comes at nosurprise and is by construction since the MV IS optimization relies on future informationwhich in reality is not actually available by the time of portfolio creation. Still, it definesan upper bound for the maximum attainable Sharpe ratio in case of perfect foresight andhence can be used to assess the severity of performance decrease based on this informationasymmetry when compared to the MV strategy. Whereas the decrease in case of the


Table 6.1: Sharpe Ratios. This table lists the out-of-sample Sharpe ratios (SRs) of the naive diversification(EW) approach, the in-sample SRs for the mean-variance (MV IS) approach, and the out-of-sample SRs ofthe 15 optimized portfolio strategies for all six data sets under consideration. The first number in parenthesesrepresents the p-value of the one-sided JK test on the pairwise difference between the out-of-sample SR ofEW and strategy i, i = MV,MinV, . . . ,RRT. The second number in parentheses represents the p-value ofthe corresponding two-sided LW bootstrap test. SRs that outperform the EW benchmark at a significancelevel of α = 0.1 with respect to LW are highlighted in bold. The numbers indicated by - · - denote therelative ranking of the specific strategy within each data set based on the value of the SR. All results relyon a rolling-window approach with window length M = 120, risk-aversion parameter γ = 1, and tuningparameter η = 1 (only VT and RRT).

Strategy PF6 USAN = 6

PF25 USAN = 25

IND10 USAN = 10

IND48 USAN = 48

PF6 EUN = 6

PF25 EUN = 25

EW 0.2396 -12- 0.2386 -12- 0.2663 -9- 0.2379 -6- 0.1379 -14- 0.1461 -11-

MV IS 0.4460 -1- 0.6351 -1- 0.3399 -1- 0.4350 -1- 0.3896 -1- 0.6324 -1-

MV 0.3912 0.3942 0.2208 0.1037 0.1861 0.2227(0.00) -6- (0.00) -7- (0.17) -13- (0.02) -14- (0.23) -8- (0.20) -4-(0.01) (0.03) (0.44) (0.09) (0.58) (0.45)

MinV 0.3964 0.4302 0.3194 0.2078 0.2219 0.1383(0.00) -4- (0.00) -5- (0.08) -3- (0.27) -8- (0.02) -2- (0.44) -14-(0.00) (0.00) (0.21) (0.56) (0.09) (0.85)

VW 0.1525 0.1525 0.1525 0.1525 0.0964 0.0964(0.00) -16- (0.00) -17- (0.00) -16- (0.00) -11- (0.00) -16- (0.00) -17-(0.00) (0.00) (0.00) (0.00) (0.00) (0.00)

BS 0.4122 0.4682 0.2827 0.1564 0.2040 0.2261(0.00) -2- (0.00) -3- (0.34) -8- (0.08) -10- (0.13) -3- (0.18) -3-(0.00) (0.00) (0.75) (0.27) (0.35) (0.41)

MVsc 0.2640 0.2658 0.1653 0.1342 0.1688 0.1627(0.08) -11- (0.08) -10- (0.00) -15- (0.00) -12- (0.06) -11- (0.18) -9-(0.21) (0.23) (0.03) (0.02) (0.28) (0.42)

MinVsc 0.2665 0.2756 0.3091 0.2823 0.1802 0.1803(0.07) -10- (0.03) -9- (0.07) -4- (0.08) -4- (0.01) -9- (0.05) -8-(0.13) (0.05) (0.20) (0.17) (0.08) (0.20)

BSsc 0.2683 0.2605 0.2010 0.1780 0.1700 0.1595(0.05) -9- (0.11) -11- (0.02) -14- (0.04) -9- (0.04) -10- (0.23) -10-(0.13) (0.33) (0.07) (0.13) (0.27) (0.50)

MVlsc 0.2793 0.2278 0.0953 0.0556 0.1874 0.2033(0.09) -8- (0.42) -13- (0.00) -17- (0.00) -15- (0.06) -7- (0.16) -7-(0.20) (0.84) (0.00) (0.02) (0.29) (0.43)

MP 0.2302 0.2270 0.2351 0.2233 0.1343 0.1420(0.00) -13- (0.00) -14- (0.00) -12- (0.00) -7- (0.00) -15- (0.00) -12-(0.03) (0.03) (0.03) (0.00) (0.03) (0.02)

MV/MinV 0.4122 0.4712 0.2630 0.0476 0.1946 0.2201(0.00) -3- (0.00) -2- (0.47) -10- (0.00) -16- (0.15) -6- (0.18) -5-(0.00) (0.00) (0.94) (0.01) (0.43) (0.40)

EW/MinV 0.3826 0.3938 0.3215 0.2648 0.1988 0.1404(0.00) -7- (0.00) -8- (0.03) -2- (0.13) -5- (0.03) -4- (0.42) -13-(0.00) (0.00) (0.08) (0.27) (0.14) (0.83)

EW/MV 0.3962 0.4086 0.2623 0.1278 0.1951 0.2337(0.00) -5- (0.00) -6- (0.41) -11- (0.00) -13- (0.10) -5- (0.10) -2-(0.00) (0.00) (0.83) (0.00) (0.30) (0.24)

EW/MV/MinV 0.0162 0.4501 0.2959 0.0026 −0.0816 0.2050(0.00) -17- (0.00) -4- (0.17) -6- (0.00) -17- (0.00) -17- (0.26) -6-(0.04) (0.00) (0.41) (0.02) (0.04) (0.56)

VT 0.2282 0.2102 0.2972 0.2886 0.1422 0.1287(0.01) -14- (0.00) -15- (0.00) -5- (0.00) -2- (0.27) -12- (0.03) -15-(0.10) (0.00) (0.03) (0.01) (0.56) (0.06)

RRT 0.2178 0.1934 0.2862 0.2879 0.1412 0.1211(0.00) -15- (0.00) -16- (0.02) -7- (0.00) -3- (0.32) -13- (0.01) -16-(0.02) (0.00) (0.10) (0.01) (0.68) (0.02)


PF6 USA portfolio is rather limited, performance drops are substantial for the remainingdata sets. The Sharpe ratio decreases, for example, from 0.64 to 0.39 for PF25 USA.Already the comparison between MV IS and MV illustrates the losses in performancedue to estimation errors and the resulting challenge of obtaining statistically significantout-of-sample outperformances relative to EW.

A second observation considers the limited number of optimized portfolio strategiesthat achieve to indeed significantly outperform EW. Out of the 90 listed optimized out-of-sample portfolio Sharpe ratios only 21 show a significant outperformance based on thetwo-sided p-value of the studentized bootstrap Ledoit and Wolf (2008) test for a signif-icance level of α = 0.1. Moreover, these few positive findings are neither attributableto a single optimized strategy nor to a specific data set. In fact, the results already nowsuggest that no optimized portfolio strategy consistently outperforms EW and confirmsthe findings of the related empirical literature which judges EW a tough benchmark tobeat. It appears as if the inherent estimation risks completely outweigh theoretically pro-posed performance gains. In case of EW/MV/MinV and PF6 EU, a negative Sharpe ratiois obtained, i.e. the performance of this particular strategy is even inferior relative to aninvestment in the risk-free rate.

The findings of these first two observations are disappointing, yet not completely un-expected. The general performance behavior of the optimized strategies is very hetero-geneous across data sets and deserves a closer examination. Although inferior to MV IS,the MV strategy substantially and significantly outperforms EW in case of PF6 and PF25USA. Focusing on PF6 USA, MV outperforms EW by a factor of 1.6. Comparably highand significant performance levels are obtained for MinV, BS, MV/MinV, EW/MinV, andEW/MV. For PF25 USA, MV/MinV even outperforms EW by a factor of more than 2.In fact, all unconstrained optimized portfolio strategies apart from VW and MP appearto significantly outperform EW for PF6 and PF25 USA. No such outperformance can befound neither for the two US industry data sets nor for the two European portfolios. TheVW strategies achieve identical levels of Sharpe ratios for the US and European data sets,respectively, since the value of the portfolios belonging to the same region is indepen-dent of the specific disposition of the portfolios. However, no superior performance isachieved across the data sets. VT and RRT are not explicitly designed as short sale con-straint strategies but effectively are since they only attain positive weights. The two timingstrategies yield significant results in case of both US industry portfolios, i.e. IND10 andIND48 USA. They are also the only strategies that consistently achieve a significantlyhigher Sharpe ratio for these two data sets. The good performance of the timing strate-gies with respect to the US industry portfolios can be related to specific characteristicsinherent in these data sets. IND48, for example, exhibits the highest dispersion in meanand variance across the included assets of all data sets under consideration. Exactly thesecharacteristics positively affect the performance of the timing strategies.

To shed some more light on the actual optimization process, Table 6.2 yields aver-


Table 6.2: Average Minimum and Maximum Sharpe Ratio Portfolio Weights. This table lists the aver-age out-of-sample minimum and maximum portfolio weights for every strategy and all six data sets underconsideration. This information is missing for the value-weighted (VW) strategy as only data about the finalportfolio returns are available but no specific corresponding portfolio weights. The numbers are obtainedby averaging the weights of every single asset over the out-of-sample estimation period. Presented, how-ever, are only the minimum and maximum values of these averages. All results rely on a rolling-windowapproach with window length M = 120, risk-aversion parameter γ = 1, and tuning parameter η = 1 (onlyVT and RRT).

PF6 USAN = 6

PF25 USAN = 25

IND10 USAN = 10

IND48 USAN = 48

PF6 EUN = 6

PF25 EUN = 25

Strategy Min. Max. Min. Max. Min. Max. Min. Max. Min. Max. Min. Max.

EW 0.17 0.17 0.04 0.04 0.10 0.10 0.02 0.02 0.17 0.17 0.04 0.04

MV −2.02 2.32 −1.53 1.04 −0.44 0.68 −0.34 0.39 −1.93 2.46 −8.37 9.27

MinV −1.02 1.17 −0.44 0.66 −0.40 0.44 −0.16 0.38 −0.53 1.44 −0.56 1.50

VW - - - - - - - - - - - -

BS −1.62 1.86 −1.11 0.88 −0.43 0.45 −0.23 0.30 −1.15 2.00 −5.68 6.81

MVsc 0.00 0.62 0.00 0.25 0.00 0.36 0.00 0.21 0.00 0.75 0.00 0.34

MinVsc 0.00 0.38 0.00 0.24 0.00 0.40 0.00 0.34 0.00 0.58 0.00 0.45

BSsc 0.00 0.59 0.00 0.20 0.00 0.41 0.00 0.21 0.00 0.70 0.00 0.30

MVlsc −0.30 0.97 −0.30 0.83 −0.28 0.59 −0.30 0.79 −0.30 0.96 −0.30 1.01

MP 0.13 0.22 0.03 0.06 0.05 0.14 0.01 0.03 0.15 0.19 0.03 0.05

MV/MinV −1.49 1.70 −0.75 0.66 −0.46 0.55 −0.33 0.53 −0.89 2.06 −3.77 5.04

EW/MinV −0.82 1.01 −0.30 0.48 −0.27 0.35 −0.06 0.19 −0.37 1.22 −0.30 0.86

EW/MV −1.31 1.64 −0.61 0.39 −0.03 0.29 −0.15 0.23 −0.75 1.85 −2.99 3.44

EW/MV/MinV −2.49 2.74 −0.63 0.59 −0.26 0.40 −0.26 0.18 −0.99 1.43 −6.06 7.49

VT 0.09 0.27 0.01 0.16 0.03 0.45 0.00 0.19 0.06 0.35 0.01 0.22

RRT 0.08 0.33 0.01 0.20 0.03 0.39 0.00 0.21 0.07 0.37 0.01 0.24

age minimum and maximum portfolio weights corresponding to the results regarding theSharpe ratio. In case of EW, the lower and upper extreme portfolio weights coincide byconstruction and represent the equal investment across all available assets. The minimumand maximum weights of MP closely resemble the corresponding EW weights which alsotranslates into similar Sharpe ratios. Estimation risk inherent in the process of determin-ing the MP weights, however, renders the performance of MP to be inferior relative toEW.

A concern often raised in the context of portfolio optimization relates to extreme port-folio weight positions that are obtained thus exposing the optimization problem to moreadverse effects of estimation errors. Michaud (1989) even argues that mean-variance op-timization can be referred to as the act of error maximization since assets with a wronglyestimated high expected return and a wrongly estimated low variance tend to be assignedlarge portfolio weights and vice versa. Consequently, the resulting portfolios exhibit anupward bias with respect to the estimate of the portfolio mean and a downward bias withrespect to the portfolio variance. However, portfolio weight positions in this analysis arelimited. The most extreme value is attained for the PF25 EU data set with respect to MVand yields an average maximum long position in an asset of approximately 930%. Apart


from the PF25 EU data set for which more extreme portfolio weight positions are real-ized, weights for the unconstrained optimized portfolio strategies hardly ever exceed theabsolute value of 250%. The latter number might represent an extreme value in real-worldtrading but is rather limited regarding the theoretical portfolio optimization literature. Aninteresting comparison can be conducted by examining MV, MVsc, and MVlsc sincethese strategies represent three different levels of constraining portfolio weights. MV isnot subject to any constraints and hence results in the most extreme portfolio weight po-sitions also relative to all other strategies. MVsc prohibits short selling completely suchthat all portfolio weights are forced to adopt at least a zero holding position. Moreover,the additional restriction of the portfolio weights to sum to one effectively leads to a longsale constraint with threshold one. Indeed, MVsc as well as the other short sale constraintstrategies only obtain weights between zero and one. In addition, lower average minimumweights frequently adopt a zero holding position, i.e. at least one asset is not included inthe respective portfolio. MVlsc is a somewhat intermediate scenario that allows shortsales but only up to a threshold of −30%. This lower bound is often attained allowingthe respective average maximum return to exceed the boundary of 100% which, however,only happens in case of PF25 EU. Although MVlsc utilizes the possibility of realizinga specific level of short selling, the resulting performance with respect to Sharpe ratiosis comparable. As far as the results of this analysis are concerned, short and long saleconstraints limit the out-of-sample Sharpe ratio performance.

Results for the Sortino and Omega ratios are listed in Table 6.3. The major drawbackof these performance evaluation criteria as well as for the Calmar ratio and the returnon value-at-risk is the lack of an appropriate statistical test to differentiate between theEW benchmark performance and the respective optimized portfolio strategies. This issueis clearly under-researched. Nevertheless, valuable information can be extracted whencomparing the results of the Sortino ratio to the Sharpe ratio. Apart from EW/MV/MinVin combination with PF6 EU, all Sortino ratios exceed the corresponding Sharpe ratios.This behavior is expected since the Sortino ratio only considers downside risk defined asnegative deviations from the minimum acceptable return of zero. However, the Sortinoratios are less than twice as high as the corresponding Sharpe ratios as would be thecase for normally distributed returns. This is again suggestive of the negative skewnessinherent in the return data such that by considering only adverse deviations from theminimum acceptable return still more than half of the volatility for the whole distributionis captured. Optimized portfolio strategies that achieve good results for the Sharpe ratioalso continue to exhibit the highest values in case of the Sortino ratio. Results for theOmega ratio are in line with previous findings. Since the Omega ratio is a nonparametricreward-to-risk ratio but achieves comparable outcomes, the loss of considering only thefirst two moments of the return distribution as for the Sharpe ratio does not seem to heavilyaffect performance behavior.

Table 6.4 presents the results for the Calmar ratios and the returns on value-at-risk. In


Table 6.3: Sortino and Omega Ratios. This table lists the out-of-sample Sortino ratios (SoRs) and Omegaratios (OR) of the naive diversification (EW) approach, the in-sample SoRs and ORs for the mean-variance(MV IS) approach, and the out-of-sample SoRs and ORs of the 15 optimized portfolio strategies for allsix data sets under consideration. The numbers indicated by - · - denote the relative ranking of the specificstrategy within each data set based on the value of the SoR and OR. All results rely on a rolling-windowapproach with window length M = 120, risk-aversion parameter γ = 1, and tuning parameter η = 1 (onlyVT and RRT).

Sortino Ratios


PF25 USAN = 25

IND10 USAN = 10

IND48 USAN = 48

PF6 EUN = 6

PF25 EUN = 25

EW 0.3622 -12- 0.3614 -12- 0.4217 -10- 0.3679 -6- 0.1977 -14- 0.2094 -11-

MV IS 0.8556 -1- 1.5820 -1- 0.5943 -1- 0.9093 -1- 0.7433 -1- 1.6951 -1-

MV 0.7372 -4- 0.7365 -7- 0.3459 -13- 0.1506 -14- 0.2793 -7- 0.4176 -2-

MinV 0.7224 -5- 0.7871 -5- 0.5575 -2- 0.3344 -8- 0.3230 -2- 0.2029 -13-

VW 0.2213 -16- 0.2213 -17- 0.2213 -16- 0.2213 -11- 0.1363 -16- 0.1363 -17-

BS 0.7827 -2- 0.9405 -3- 0.4695 -7- 0.2418 -10- 0.3001 -3- 0.4164 -3-

MVsc 0.3987 -11- 0.4032 -10- 0.2464 -15- 0.1992 -13- 0.2456 -11- 0.2360 -9-

MinVsc 0.4183 -9- 0.4366 -9- 0.5270 -4- 0.4633 -2- 0.2601 -9- 0.2606 -8-

BSsc 0.4058 -10- 0.3952 -11- 0.3092 -14- 0.2693 -9- 0.2473 -10- 0.2306 -10-

MVlsc 0.4413 -8- 0.3608 -13- 0.1401 -17- 0.0813 -15- 0.2746 -8- 0.3162 -7-

MP 0.3471 -13- 0.3433 -14- 0.3651 -12- 0.3442 -7- 0.1924 -15- 0.2033 -12-

MV/MinV 0.7827 -3- 0.9533 -2- 0.4360 -9- 0.0733 -16- 0.2850 -5- 0.3879 -5-

EW/MinV 0.6692 -7- 0.6713 -8- 0.5469 -3- 0.4138 -5- 0.2847 -6- 0.1975 -14-

EW/MV 0.7052 -6- 0.7684 -6- 0.4194 -11- 0.2147 -12- 0.2862 -4- 0.4083 -4-

EW/MV/MinV 0.0171 -17- 0.8476 -4- 0.4993 -5- 0.0028 -17- −0.0855 -17- 0.3805 -6-

VT 0.3448 -14- 0.3182 -15- 0.4809 -6- 0.4585 -4- 0.2023 -12- 0.1816 -15-

RRT 0.3288 -15- 0.2938 -16- 0.4601 -8- 0.4605 -3- 0.2009 -13- 0.1705 -16-Omega Ratios


PF25 USAN = 25

IND10 USAN = 10

IND48 USAN = 48

PF6 EUN = 6

PF25 EUN = 25

EW 1.8590 -12- 1.8548 -12- 2.0080 -9- 1.8789 -6- 1.4411 -14- 1.4752 -11-

MV IS 3.1504 -1- 5.5764 -1- 2.4311 -1- 3.3102 -1- 2.9362 -1- 5.6607 -1-

MV 2.7561 -5- 2.8840 -7- 1.7662 -13- 1.3334 -14- 1.6787 -7- 1.9233 -2-

MinV 2.7442 -6- 3.0296 -5- 2.2574 -3- 1.7156 -8- 1.7916 -2- 1.4648 -13-

VW 1.4856 -16- 1.4856 -17- 1.4856 -16- 1.4856 -12- 1.2906 -16- 1.2906 -17-

BS 2.8822 -3- 3.4604 -3- 2.0461 -8- 1.5318 -10- 1.7473 -3- 1.9011 -3-

MVsc 2.0111 -10- 2.0047 -10- 1.5698 -15- 1.4435 -13- 1.5588 -11- 1.5356 -9-

MinVsc 2.0010 -11- 2.0543 -9- 2.2244 -4- 2.0921 -4- 1.6131 -9- 1.6161 -8-

BSsc 2.0364 -9- 1.9698 -11- 1.7340 -14- 1.6136 -9- 1.5703 -10- 1.5256 -10-

MVlsc 2.1317 -8- 1.8301 -13- 1.2965 -17- 1.1646 -15- 1.6498 -8- 1.7287 -7-

MP 1.8124 -13- 1.7989 -14- 1.8536 -12- 1.8106 -7- 1.4272 -15- 1.4586 -14-

MV/MinV 2.8827 -2- 3.5023 -2- 1.9621 -11- 1.1474 -16- 1.7136 -5- 1.8382 -6-

EW/MinV 2.6660 -7- 2.8057 -8- 2.2876 -2- 2.0001 -5- 1.6902 -6- 1.4662 -12-

EW/MV 2.7933 -4- 3.0148 -6- 1.9689 -10- 1.5061 -11- 1.7175 -4- 1.8939 -4-

EW/MV/MinV 1.1538 -17- 3.3063 -4- 2.1387 -6- 1.0143 -17- 0.5361 -17- 1.8543 -5-

VT 1.8023 -14- 1.7277 -15- 2.1984 -5- 2.1654 -2- 1.4620 -12- 1.4093 -15-

RRT 1.7527 -15- 1.6588 -16- 2.1339 -7- 2.1591 -3- 1.4571 -13- 1.3802 -16-


accordance to the previous reward-to-risk ratios, unconstrained strategies related to thesample-based mean-variance and minimum-variance strategy as well as BS exhibit highvalues. VT and RRT continue to achieve high relative rankings for the two US industrydata sets.

Results for the certainty equivalent returns are listed in Table 6.5. Since the certaintyequivalent return is a common performance evaluation criterion, statistical tests to decideon the pairwise difference between the EW benchmark performance and the respectiveoptimized strategies are available. Typically, the outperformance is decided on the Deltamethod as in DeMiguel et al. (2009b). However, this analysis additionally adjusted theSharpe ratio studentized bootstrap methodology as introduced in Ledoit and Wolf (2008)to the context of certainty equivalent returns. This more general test allows for statisticalinferences that do not crucially depend on the normality assumption of return distribu-tions and hence allows for more reliable conclusions. Indeed, all out-of-sample portfolioreturn time series are statistically different from a normal distribution at a significancelevel of α = 0.01 based on the Jarque-Bera test (results are not reported) and hence justifythe use of a test statistic which accounts for this fact. As a result, a total of 17 certaintyequivalent returns are superior relative to the EW benchmark of which 16 are related toeither the PF6 or PF25 USA data sets. Superior optimized portfolio strategies are simi-lar to the Sharpe ratio setting and are mainly MV, MinV, BS, MV/MinV, EW/MinV, andEW/MV. The main result, however, is unchanged and summarizes in the lack of an opti-mized portfolio strategy that achieves to consistently outperform the naive diversificationbenchmark. An explanation of the good performances of EW can be found when re-considering the underlying data sets. All assets included in the data sets are obtained bycombining several individual assets into portfolios such that the final data set can be betterdescribed as a portfolio of portfolios. Hence, investing naively in a selection of portfoliostypically leads to better results relative to naively investing in a selection of individualassets that exhibit a high level of idiosyncratic volatility. The differences between the USand European PF6 and PF25 portfolios with respect to optimized portfolio strategies thatachieve significant outperformances might be attributed to the different underlying timeperiod available for estimation. The probability of the optimized strategies to outperformEW increases with an increasing out-of-sample estimation period as estimation precisionincreases and hence the European data sets face a greater challenge due to their relativelylimited number of observations.

As a final step and to somehow circumvent the shortcoming of not having reliable sta-tistical test statistics to differentiate between the reward-to-risk ratios of Sortino, Omega,Calmar, and the return on value-at-risk, a rank correlation investigation is applied in thestyle of Eling (2008). Table 6.6 presents the rank correlations with respect to the perfor-mance evaluation criteria for the two US and European PF6 and PF25 data sets. In case ofPF6 USA, the lowest rank correlation still amounts to a value of 0.95 between SR and CRas well as CR and RoVaR. Equally high and significant correlations are obtained for PF25


Table 6.4: Calmar Ratios and Returns on Value-at-Risk. This table lists the out-of-sample Calmar ratios(CRs) and returns on value-at-risk (RoVaR) of the naive diversification (EW) approach, the in-sample CRsand RoVaRs for the mean-variance (MV IS) approach, and the out-of-sample CRs and RoVaRs of the 15optimized portfolio strategies for all six data sets under consideration. The numbers indicated by - · - denotethe relative ranking of the specific strategy within each data set based on the value of the CR and RoVaR.All results rely on a rolling-window approach with window length M = 120, risk-aversion parameter γ = 1,and tuning parameter η = 1 (only VT and RRT).

Calmar Ratios


PF25 USAN = 25

IND10 USAN = 10

IND48 USAN = 48

PF6 EUN = 6

PF25 EUN = 25

EW 0.0209 -14- 0.0215 -13- 0.0226 -11- 0.0207 -6- 0.0124 -14- 0.0131 -11-

MV IS 0.0883 -1- 0.1711 -1- 0.0482 -1- 0.0684 -1- 0.0579 -1- 0.1682 -1-

MV 0.0805 -2- 0.0404 -7- 0.0252 -10- 0.0120 -14- 0.0227 -2- 0.0639 -2-

MinV 0.0473 -6- 0.0487 -6- 0.0382 -2- 0.0197 -7- 0.0208 -4- 0.0118 -13-

VW 0.0124 -16- 0.0124 -17- 0.0124 -16- 0.0124 -13- 0.0086 -16- 0.0086 -17-

BS 0.0754 -3- 0.0825 -2- 0.0345 -4- 0.0132 -11- 0.0220 -3- 0.0566 -3-

MVsc 0.0223 -10- 0.0226 -10- 0.0131 -15- 0.0131 -12- 0.0155 -10- 0.0147 -9-

MinVsc 0.0218 -11- 0.0225 -11- 0.0312 -6- 0.0255 -4- 0.0156 -9- 0.0153 -8-

BSsc 0.0225 -9- 0.0224 -12- 0.0167 -14- 0.0176 -9- 0.0154 -11- 0.0143 -10-

MVlsc 0.0271 -8- 0.0294 -9- 0.0097 -17- 0.0119 -15- 0.0181 -7- 0.0238 -7-

MP 0.0208 -15- 0.0211 -14- 0.0185 -13- 0.0195 -8- 0.0121 -15- 0.0128 -12-

MV/MinV 0.0668 -4- 0.0794 -3- 0.0314 -5- 0.0058 -16- 0.0205 -5- 0.0415 -5-

EW/MinV 0.0408 -7- 0.0381 -8- 0.0376 -3- 0.0220 -5- 0.0178 -8- 0.0117 -14-

EW/MV 0.0512 -5- 0.0659 -5- 0.0218 -12- 0.0142 -10- 0.0198 -6- 0.0356 -6-

EW/MV/MinV 0.0007 -17- 0.0697 -4- 0.0281 -7- 0.0001 -17- −0.0414 -17- 0.0489 -4-

VT 0.0212 -12- 0.0203 -15- 0.0273 -8- 0.0285 -3- 0.0127 -12- 0.0112 -15-

RRT 0.0210 -13- 0.0193 -16- 0.0261 -9- 0.0294 -2- 0.0126 -13- 0.0106 -16-Returns on Value-at-Risk


PF25 USAN = 25

IND10 USAN = 10

IND48 USAN = 48

PF6 EUN = 6

PF25 EUN = 25

EW 0.1705 -12- 0.1696 -12- 0.1932 -9- 0.1690 -6- 0.0915 -14- 0.0975 -11-

MV IS 0.3760 -1- 0.6602 -1- 0.2645 -1- 0.3888 -1- 0.3176 -1- 0.7000 -1-

MV 0.3120 -6- 0.3152 -7- 0.1550 -13- 0.0673 -14- 0.1276 -8- 0.1566 -4-

MinV 0.3176 -4- 0.3542 -5- 0.2410 -3- 0.1446 -8- 0.1560 -2- 0.0918 -14-

VW 0.1022 -16- 0.1022 -17- 0.1022 -16- 0.1022 -11- 0.0623 -16- 0.0623 -17-

BS 0.3344 -2- 0.3979 -3- 0.2075 -8- 0.1051 -10- 0.1416 -3- 0.1594 -3-

MVsc 0.1912 -11- 0.1927 -10- 0.1117 -15- 0.0889 -12- 0.1143 -11- 0.1097 -9-

MinVsc 0.1933 -10- 0.2012 -9- 0.2314 -4- 0.2072 -4- 0.1231 -9- 0.1231 -8-

BSsc 0.1949 -9- 0.1882 -11- 0.1392 -14- 0.1213 -9- 0.1153 -10- 0.1074 -10-

MVlsc 0.2046 -8- 0.1608 -13- 0.0615 -17- 0.0350 -15- 0.1285 -7- 0.1410 -7-

MP 0.1627 -13- 0.1601 -14- 0.1667 -12- 0.1571 -7- 0.0889 -15- 0.0945 -12-

MV/MinV 0.3344 -3- 0.4015 -2- 0.1903 -10- 0.0298 -16- 0.1342 -6- 0.1545 -5-

EW/MinV 0.3031 -7- 0.3148 -8- 0.2429 -2- 0.1919 -5- 0.1375 -4- 0.0933 -13-

EW/MV 0.3173 -5- 0.3305 -6- 0.1897 -11- 0.0843 -13- 0.1346 -5- 0.1656 -2-

EW/MV/MinV 0.0099 -17- 0.3767 -4- 0.2194 -6- 0.0016 -17- −0.0473 -17- 0.1424 -6-

VT 0.1611 -14- 0.1465 -15- 0.2205 -5- 0.2128 -2- 0.0947 -12- 0.0849 -15-

RRT 0.1526 -15- 0.1333 -16- 0.2106 -7- 0.2122 -3- 0.0939 -13- 0.0795 -16-


Table 6.5: Certainty Equivalent Returns. This table lists the out-of-sample certainty equivalent returns(CEs) of the naive diversification (EW) approach, the in-sample CEs for the mean-variance (MV IS) ap-proach, and the out-of-sample CEs of the 15 optimized portfolio strategies for all six data sets under con-sideration. The first number in parentheses represents the p-value of the one-sided JK test on the pairwisedifference between the out-of-sample CE of EW and strategy i, i = MV,MinV, . . . ,RRT. The secondnumber in parentheses represents the p-value of the corresponding two-sided LW bootstrap test. CEs thatoutperform the EW benchmark at a significance level of α = 0.1 with respect to LW are highlighted inbold. The numbers indicated by - · - denote the relative ranking of the specific strategy within each data setbased on the value of the CE. All results rely on a rolling-window approach with window length M = 120,risk-aversion parameter γ = 1, and tuning parameter η = 1 (only VT and RRT).


PF25 USAN = 25

IND10 USAN = 10

IND48 USAN = 48

PF6 EUN = 6

PF25 EUN = 25

EW 0.0102 -12- 0.0104 -13- 0.0101 -6- 0.0099 -4- 0.0060 -14- 0.0065 -11-

MV IS 0.0196 -1- 0.0281 -1- 0.0114 -1- 0.0173 -1- 0.0285 -1- 0.0900 -1-

MV 0.0184 0.0228 0.0093 0.0053 0.0106 0.0248(0.00) -2- (0.00) -2- (0.35) -13- (0.20) -14- (0.14) -2- (0.12) -2-(0.00) (0.00) (0.72) (0.47) (0.36) (0.20)

MinV 0.0145 0.0154 0.0105 0.0076 0.0094 0.0057(0.00) -7- (0.00) -8- (0.39) -4- (0.13) -11- (0.06) -7- (0.38) -14-(0.02) (0.01) (0.81) (0.30) (0.16) (0.75)

VW 0.0057 0.0057 0.0057 0.0057 0.0037 0.0037(0.00) -16- (0.00) -17- (0.00) -16- (0.00) -13- (0.00) -16- (0.00) -17-(0.00) (0.00) (0.00) (0.00) (0.00) (0.00)

BS 0.0169 0.0210 0.0099 0.0079 0.0102 0.0228(0.00) -3- (0.00) -3- (0.46) -9- (0.26) -9- (0.10) -3- (0.06) -3-(0.01) (0.00) (0.94) (0.59) (0.29) (0.12)

MVsc 0.0120 0.0119 0.0082 0.0072 0.0079 0.0077(0.02) -10- (0.05) -10- (0.17) -15- (0.14) -12- (0.04) -10- (0.12) -9-(0.04) (0.17) (0.35) (0.36) (0.22) (0.30)

MinVsc 0.0104 0.0105 0.0101 0.0093 0.0078 0.0077(0.40) -11- (0.45) -12- (0.50) -7- (0.33) -6- (0.03) -11- (0.15) -8-(0.81) (0.91) (0.99) (0.67) (0.12) (0.41)

BSsc 0.0121 0.0116 0.0090 0.0086 0.0079 0.0074(0.01) -9- (0.10) -11- (0.24) -14- (0.23) -8- (0.03) -9- (0.18) -10-(0.03) (0.29) (0.46) (0.54) (0.19) (0.42)

MVlsc 0.0146 0.0167 0.0045 −0.0109 0.0096 0.0149(0.00) -6- (0.06) -7- (0.07) -17- (0.02) -16- (0.03) -4- (0.05) -7-(0.01) (0.14) (0.13) (0.07) (0.22) (0.25)

MP 0.0100 0.0102 0.0094 0.0098 0.0058 0.0063(0.19) -13- (0.09) -14- (0.09) -12- (0.24) -5- (0.00) -15- (0.00) -12-(0.39) (0.19) (0.20) (0.54) (0.03) (0.01)

MV/MinV 0.0165 0.0198 0.0095 −0.0012 0.0095 0.0182(0.00) -4- (0.00) -4- (0.37) -11- (0.02) -15- (0.13) -6- (0.06) -5-(0.00) (0.00) (0.76) (0.03) (0.38) (0.13)

EW/MinV 0.0138 0.0139 0.0104 0.0092 0.0084 0.0057(0.00) -8- (0.00) -9- (0.38) -5- (0.25) -7- (0.08) -8- (0.29) -13-(0.01) (0.01) (0.80) (0.54) (0.22) (0.56)

EW/MV 0.0163 0.0183 0.0098 0.0076 0.0096 0.0171(0.00) -5- (0.00) -5- (0.33) -10- (0.25) -10- (0.08) -5- (0.03) -6-(0.00) (0.00) (0.67) (0.41) (0.27) (0.07)

EW/MV/MinV −0.0582 0.0178 0.0099 −0.0152 −0.1815 0.0203(0.00) -17- (0.00) -6- (0.46) -8- (0.00) -17- (0.00) -17- (0.13) -4-(0.45) (0.00) (0.93) (0.38) (0.41) (0.24)

VT 0.0100 0.0097 0.0107 0.0108 0.0062 0.0055(0.25) -14- (0.08) -15- (0.07) -2- (0.14) -3- (0.27) -12- (0.02) -15-(0.55) (0.20) (0.19) (0.29) (0.57) (0.04)

RRT 0.0098 0.0092 0.0106 0.0110 0.0062 0.0051(0.18) -15- (0.05) -16- (0.07) -3- (0.09) -2- (0.31) -13- (0.00) -16-(0.35) (0.09) (0.16) (0.18) (0.66) (0.01)


Table 6.6: Performance Evaluation Criteria Rank Correlations. This table lists the Spearman’s rankcorrelation coefficients for all performance evaluation criteria with respect to the data sets PF6 USA, PF25USA, PF6 EU, and PF25 EU. Hence, this table contains a total of four correlation matrices. The lower lefttriangular matrix of the first group column denoted by PF6 / PF25 USA containing plain numbers representsthe correlation matrix of the PF6 USA data set. The upper right triangular matrix of the first group columndenoted by PF6 / PF25 USA containing italic numbers represents the correlation matrix of the PF25 USAdata set. The second group column is defined accordingly. All listed rank correlations are significantlydifferent from zero for a given significance level of α = 0.01.

Performance PF6 / PF25 USA PF6 / PF25 EUMeasure SR SoR OR CR RoVaR CE SR SoR OR CR RoVaR CE

SR 1.00 1.00 1.00 0.97 1.00 0.89 1.00 0.99 0.98 0.97 1.00 0.97SoR 0.99 1.00 1.00 0.97 1.00 0.89 0.99 1.00 0.99 0.99 0.99 0.99OR 0.99 0.98 1.00 0.97 1.00 0.89 0.99 1.00 1.00 0.98 0.98 0.99CR 0.95 0.96 0.96 1.00 0.97 0.95 0.93 0.95 0.95 1.00 0.97 1.00RoVaR 1.00 0.99 0.99 0.95 1.00 0.89 1.00 0.99 0.99 0.93 1.00 0.97CE 0.96 0.97 0.98 0.97 0.96 1.00 0.89 0.90 0.90 0.97 0.89 1.00

USA, PF6 EU, and PF25 EU. As a result, the choice of a performance evaluation criteriondoes not excessively alter the relative performance ranking of the portfolio strategies un-der consideration if at all. Therefore, the Sharpe ratio appears to be a valid performanceevaluation criterion despite its reliance on normally distributed return data.

The rank correlations listed in Table 6.7 focus on the relations between the six datasets under consideration. This time, rank correlations exhibit very heterogeneous values.In case of the Sharpe ratio, the correlation between PF6 USA and PF6 EU is high andamounts to 0.94. However, even the correlations between PF6 USA and PF25 USA butalso between PF25 USA and PF25 EU decrease to 0.74 and 0.77, respectively, and henceare already lower than the lowest performance evaluation criteria rank correlation of 0.89.

Table 6.7: Data Sets Rank Correlations. This table lists the Spearman’s rank correlation coefficients forall data sets with respect to the Sharpe ratio and certainty equivalent returns. Hence, this table containsa total of two correlation matrices. The lower left triangular matrix containing plain numbers representsthe correlation matrix of the Sharpe ratios. The upper right triangular matrix containing italic numbersrepresents the correlation matrix of the certainty equivalent returns. The numbers in parentheses denote thep-values of a Student’s t-distributed test statistic checking for a rank correlation significantly different fromzero.

Performance Sharpe Ratio / Certainty Equivalent ReturnsMeasure PF6 USA PF25 USA IND10 USA IND48 USA PF6 EU PF25 EU

PF6 USA 1.00 0.84 −0.03 −0.10 0.97 0.68(0.00) (0.91) (0.69) (0.00) (0.00)

PF25 USA 0.74 1.00 0.00 −0.36 0.80 0.90(0.00) (1.00) (0.16) (0.00) (0.00)

IND10 USA 0.23 0.43 1.00 0.68 0.00 −0.11(0.38) (0.09) (0.00) (1.00) (0.67)

IND48 USA −0.04 −0.26 0.61 1.00 −0.09 −0.30(0.87) (0.32) (0.01) (0.74) (0.24)

PF6 EU 0.94 0.66 0.38 0.11 1.00 0.63(0.00) (0.00) (0.13) (0.69) (0.01)

PF25 EU 0.64 0.77 0.01 −0.38 0.49 1.00(0.01) (0.00) (0.97) (0.13) (0.05)

7. Robustness Checks 46

The two industry portfolios achieve an intermediate correlation of 0.61 between eachother but have very low or even negative yet insignificant rank correlations to other datasets. Similar characteristics are obtained with respect to the certainty equivalent returns.The most striking performance differences between the data sets can be attributed to theVT and RRT strategies. Typically ranging at the end of the relative evaluation for the USand European PF6 and PF25 portfolios, the two timing strategies occupy top ranks forthe two US industry data sets. Both strategies significantly outperform EW in case of theSharpe ratio but marginally fail to do so for the certainty equivalent returns. These find-ings demonstrate that the relative performance of optimized portfolio strategies cruciallydepends on the underlying data set.

7. Robustness Checks

This section presents robustness checks for the most critical underlying assumptions withrespect to the estimation methodology as well as for the most prominent parameters. Dueto limitations regarding the scope of this analysis, there is no investigation about the ef-fects of considering different subperiods on the out-of-sample portfolio performance. Aninteresting question in this context, for example, is the validity of results when excludingthe global financial crisis in 2007 and 2008 and its adverse effects on returns. All find-ings of this section only rely on the Sharpe ratio and the certainty equivalent return asperformance evaluation criteria and refer to the tables listed in Appendix D for the sakeof readability.

7.1. Estimation Window

The robustness checks related to the estimation window are threefold. First, a rolling-window approach with a shorter window length of M = 60 months is considered. Second,a rolling-window approach with a longer window length of M = 180 months is examined.Third, the rolling-window approach is substituted with an expanding-window approach.The latter estimation methodology is similar to the rolling-window approach, but insteadof always dropping the earliest return observation of the current window, all returns arekept such that the window expands through the data set.

In case of a shorter rolling-window estimation, results change both quantitatively andqualitatively for the Sharpe ratio. Only the MinV for PF6 USA and EW/MinV for PF6USA and PF25 USA continue to outperform EW. The scenario for VT and RRT regardingthe two US industry data sets is unchanged, i.e. these two strategies are still generally su-perior relative to EW. The most striking change can be seen for the two European data setsfor which MVsc, MinVsc, BSsc, and MVlsc now significantly outperform EW. In gen-eral, the reduced estimation period increases the possibility of estimation errors which inturn might lead to more extreme optimal portfolio weight positions. Indeed, the most ex-treme position in this setting is the average minimum portfolio weight for EW/MV/MinV

7. Robustness Checks 47

in case of PF25 USA which adopts a value of approximately −63, 000% (not reported).Changes with respect to the certainty equivalent returns are in line with those reported forthe Sharpe ratio.

Kritzman et al. (2010) argue that past empirical findings that speak in favor of the naivediversification approach heavily rely on the use of a short-term rolling-window estimationapproach which might not be the appropriate technique to compare different optimizedstrategies. According to Kritzman et al. (2010), the reliance on a time series of only 60or 120 months of return data as basis for estimating expected returns is not in accordancewith the assumption of a rational investor since this short window does not suffice fora reliable estimation of the data generating parameters. Following their argument, anextension of the rolling-window by five years is considered such that a total of M = 180months are included. In general, the results confirm the findings of the baseline scenario.Optimal portfolio strategies that significantly outperform EW are MV, MinV, BS, MinVsc,MV/MinV, EW/MinV, and EW/MV in case of PF6 and PF25 USA. The two US industrydata sets are characterized by an outperformance of VT and RRT in three of the four cases.The European data sets do not include a superior strategy. The results for the certaintyequivalent returns are comparable to the baseline setting. The robustness check regardingan expanding-window approach verifies the baseline results for both the Sharpe ratio andthe certainty equivalent returns. In case of the latter performance criteria, also the PF6EU data set exhibits now some outperforming optimized strategies that mainly coincidewith the superior strategies of the PF6 USA data set.

7.2. Risk-Aversion Parameter

The robustness checks for the risk-aversion parameter are realized for two different sce-narios with γ = 2 and γ = 4, respectively. The out-of-sample performances for theSharpe ratios are independent of the specific risk-aversion parameter by the majority.Only EW/MV and EW/MV/MinV are affected by the risk-aversion parameter, however,all baseline results are still valid. In contrast, the certainty equivalent returns negativelydepend on the corresponding risk-aversion parameter, i.e. the larger γ the lower the cer-tainty equivalent returns. Still, the results are almost identical relative to the baselinescenario. The only major difference occurs in case of γ = 4 for which VT and RRTachieve to outperform EW for the two US industry data sets for the first time with respectto the certainty equivalent returns.

7.3. Tuning Parameter

The so-called tuning parameter refers only to VT and RRT and can be understood as asensitivity of the resulting portfolio weights regarding changes in volatility of the returndata. The robustness checks consider values of η = 2 and η = 4, respectively. Allsuperior performances of the baseline setting remain valid. Furthermore, in case of η = 4,

8. Conclusion 48

VT and RRT also outperform EW in case of PF6 USA and PF6 EU with respect to theSharpe ratio. In the context of certainty equivalent returns, PF6 EU exhibits superiorperformances for VT and RRT for η = 2 as well as η = 4. These findings are suggestive ofan increasing performance of these timing strategies with increasing values of the tuningparameter, i.e. a higher sensitivity to volatility changes of the underlying return data.

8. Conclusion

This analysis conducts a horse race between 15 optimized portfolio strategies based onsix performance evaluation criteria across six data sets. Although no optimized portfoliostrategy can be identified that consistently outperforms the naive diversification approachacross all data sets, this finding does not automatically translate into a negation of thequestion of whether optimized portfolios can beat 1/N. In fact, several different optimizedstrategies achieve statistically superior results relative to the naive diversification approachsuch as the volatility and reward-to-risk timing strategies in case of the two US industrydata sets. Whereas the specific choice of a performance evaluation criterion seems to haveno effect on portfolio performance, results critically depend on the underlying data set.Not only do data sets that consist of a higher number of observations tend to facilitateoutperforming behavior among optimized strategies, but also other characteristics inher-ent in the data sets seem to be differently preferred by portfolio strategies. For example,the timing strategies tend to be more successful relative to other optimized strategies themore volatile the underlying data. A direction to pursue for future research is to capturethese portfolio strategy sensitivities to data set characteristics and develop a classifica-tion that links optimal strategies to data characteristics that favor a statistically significantoutperforming behavior relative to the naive diversification approach.

Appendix A. Details of the Data Sets 49

Appendix A. Details of the Data Sets

The data used for this analysis are obtained from Kenneth R. French’s website5. Thisappendix is a description of how the specific portfolios are created and further specifiesthe actual sources of the monthly return data. The idea of forming these portfolios as wellas the actual creation of them is solely attributed and performed by Kenneth R. French.The following is an excerpt of the raw description of the data generating process providedon Kenneth R. French’s website. Only minor changes have been made to the originalwording so as to fit more to the background of this analysis. The creation of the differentportfolios crucially depends on the following variables.

Market equity (size) is price times shares outstanding. Price is from ... [theCenter for Research and Security Prices (CRSP)], shares outstanding are fromCompustat (if available) or CRSP.

Book equity [(BE)] is constructed from Compustat data or collected from ...Moody’s [...]. BE is the book value of stockholders’ equity, plus balance sheetdeferred taxes and investment tax credit (if available), minus the book valueof preferred stock. Depending on availability, ... the redemption, liquidation,or par value (in that order) [is used] to estimate the book value of preferredstock. Stockholders’ equity is the value reported by Moody’s or Compustat,if it is available. If not, ... stockholders’ equity [is measured] as the bookvalue of common equity plus the par value of preferred stock, or the bookvalue of assets minus total liabilities (in that order).

A.1. US MKT, PF6, PF25, IND10, and IND48

... [MKT], the excess return on the market, [is the value-weighted] ... returnof all CRSP firms incorporated in the US and listed on the NYSE, AMEX, orNASDAQ that have a CRSP share code of 10 or 11 at the beginning of montht, good shares and price data at the beginning of t, and good return data for t

minus the one-month Treasury bill rate (from Ibbotson Associates).

The [six] portfolios, which are constructed at the end of each June, are theintersections of 2 portfolios formed on size (market equity, ME) and 3 port-folios formed on the ratio of book equity to market equity (BE/ME). The sizebreakpoint for year t is the median NYSE market equity at the end of June ofyear t. BE/ME for June of year t is the book equity for the last fiscal year endin t − 1 divided by ME for December of t − 1. The BE/ME breakpoints arethe 30th and 70th NYSE percentiles.

The [25] portfolios, which are constructed at the end of each June, are theintersections of 5 portfolios formed on size (market equity, ME) and 5 port-

5http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html


Appendix A. Details of the Data Sets 50

folios formed on the ratio of book equity to market equity (BE/ME). The sizebreakpoints for year t are the NYSE market equity quintiles at the end of Juneof t. BE/ME for June of year t is the book equity for the last fiscal year endin t − 1 divided by ME for December of t − 1. The BE/ME breakpoints areNYSE quintiles.

... [Each] NYSE, AMEX, and NASDAQ stock [is assigned] to an industryportfolio at the end of June of year t based on its four-digit [Standard Indus-trial Classification (SIC)] ... code at that time. (... Compustat SIC codes forthe fiscal year ending in calendar year t − 1 [are used]. Whenever CompustatSIC codes are not available, ... CRSP SIC codes for June of year t [are used].)... [Returns are then computed] from July of t to June of t + 1.

The portfolios for July of year t to June of t + 1 include all NYSE, AMEX,and NASDAQ stocks for which ... [there is] market equity data for Decemberof t − 1 and June of t, and (positive) book equity data for t − 1.

The ten industry sample consists of the following industries: Consumer Nondurables,Consumer Durables, Manufacturing, Energy, Business Equipment, Telecommunication,Wholesale and Retail, Health, Utilities, and Other.

The 48 industry sample consists of the following industries: Agriculture, Food Prod-ucts, Candy and Soda, Beer and Liquor, Tobacco Products, Toys, Entertainment, Printingand Publishing, Consumer Goods, Clothes, Healthcare, Medical Equipment, Pharmaceu-tical Products, Chemicals, Rubber and Plastic Products, Textiles, Construction Material,Construction, Steel, Fabricated Products, Machinery, Electrical Equipment, Automobilesand Trucks, Aircraft, Shipbuilding and Railroad Equipment, Defense, Precious Metals,Nonmetallic and Industrial Metal Mining, Coal, Petroleum and Natural Gas, Utilities,Communication, Personal Services, Business Services, Computers, Electronic Equip-ment, Measuring and Control Equipment, Business Supplies, Shipping Containers, Trans-portation, Wholesale, Retail, Restaurants and Hotels and Motels, Banking, Insurance,Real Estate, Trading, and Other.

A.2. European MKT, PF6, and PF25

The returns on the European portfolio are constructed by averaging the returns on thecountry portfolios in accordance with their Europe, Australasia, and Far East weight. Theraw data are from MSCI from 1991 to 2006 and from Bloomberg from 2007 to present.

All returns are in U.S. dollars, include dividends and capital gains, and arenot continuously compounded. ... [MKT] is the return ... [of Europe’s value-weighted] market portfolio minus the U.S. one month T-bill rate.

The [six] portfolios, which are constructed at the end of each June, are theintersections of 2 portfolios formed on size (market equity, ME) and 3 port-folios formed on the ratio of book equity to market equity (BE/ME). Big

Appendix B. Mean-Variance Optimization in Excess Returns 51

stocks are those in the top 90% of June market cap ..., and small stocks arethose in the bottom 10%. The B/M breakpoints for big and small stocks ...are the 30th and 70th percentiles of B/M for the big stocks ....

The [25] portfolios, which are constructed at the end of each June, are theintersections of 5 portfolios formed on size (market equity, ME) and 5 port-folios formed on the ratio of book equity to market equity (BE/ME). The sizebreakpoints ... are the 3rd, 7th, 13th, and 25th percentiles of ... [Europe’s ]aggregate market capitalization. The B/M breakpoints for all stocks ... are the20th, 40th, 60th, and 80th percentiles of B/M for the big (top 90% of marketcap) stocks ....

The portfolios for July of year t to June of t + 1 include all stocks for which... [there is] market equity data for December of t − 1 and June of t, and(positive) book equity data for t − 1.

Countries included in the European sample are: Austria, Belgium, Switzerland, Ger-many, Denmark, Spain, Finland, France, Great Britain, Greece, Ireland, Italy, Nether-lands, Norway, Portugal, and Sweden.

Appendix B. Mean-Variance Optimization in ExcessReturns

Letmax

xtx f ,tr f + xT

t µt −γ

2xT

t Σtxt subject to x f ,t + 1TN xt = 1 (B.1)

denote the mean-variance optimization problem in terms of nonexcess returns, i.e. µt isa vector containing the expected nonexcess returns of the N available risky assets. It ispossible to rewrite x f ,tr f + xT

t µt such that

x f ,tr f + xTt µt = x f ,tr f + xT

t (µt + 1Nr f )

= x f ,tr f + xTt µt + xT

t 1Nr f

= (x f ,t + xTt 1N)r f + xT

t µt

= r f + xTt µt. (B.2)

The first step utilizes the expression for the expected excess returns, i.e. µt = µt − 1Nr f .Hence, Eq. (B.1) can be expressed in terms of excess returns µt such that the optimizationproblem transforms to

maxxt

r f + xTt µt −

γ

2xT

t Σtxt. (B.3)

Appendix C. Minimum-Variance Optimization 52

Since the risk-free rate r f is a constant it does not alter the solution of the optimizationproblem and thus can be ignored yielding

maxxt

xTt µt −

γ

2xT

t Σtxt. (B.4)

Appendix C. Minimum-Variance Optimization

The minimum-variance strategy solves the optimization given by

minωt

12ωT

t Σtωt subject to 1TNωt = 1. (C.5)

The Lagrangian is

L(ωt, λ) =12ωT

t Σtωt − λ(1TNωt − 1). (C.6)

The first derivative with respect to the weight vector yields

∂L(ωt, λ)∂ωt

= Σtωt − λ1N = 0 (C.7)

and the first derivative with respect to the Lagrange parameter yields

∂L(ωt, λ)∂λ

= 1TNωt − 1 = 0. (C.8)

Solving Eq. (C.7) for ωt gives ωt = λΣ−1t 1N . Inserting this result in Eq. (C.8) and solving

for λ produces λ = (1TNΣ−1

t 1N)−1 with (Σ−1t )T = Σ−1

t due to symmetry of the variance-covariance matrix. Hence

ωt =Σ−1

t 1N

1TNΣ−1

t 1N. (C.9)


Appendix D. Robustness Checks Tables

Table D.1: Sharpe Ratios, Estimation Window M = 60. This table lists the out-of-sample Sharpe ra-tios (SRs) of the naive diversification (EW) approach, the in-sample SRs for the mean-variance (MV IS)approach, and the out-of-sample SRs of the 15 optimized portfolio strategies for all six data sets underconsideration. The first number in parentheses represents the p-value of the one-sided JK test on the pair-wise difference between the out-of-sample SR of EW and strategy i, i = MV,MinV, . . . ,RRT. The secondnumber in parentheses represents the p-value of the corresponding two-sided LW bootstrap test. SRs thatoutperform the EW benchmark at a significance level of α = 0.1 with respect to LW are highlighted inbold. The numbers indicated by - · - denote the relative ranking of the specific strategy within each data setbased on the value of the SR. All results rely on a rolling-window approach with window length M = 60,risk-aversion parameter γ = 1, and tuning parameter η = 1 (only VT and RRT).


PF25 USAN = 25

IND10 USAN = 10

IND48 USAN = 48

PF6 EUN = 6

PF25 EUN = 25

EW 0.2616 -10- 0.2615 -6- 0.2744 -7- 0.2530 -7- 0.1619 -9- 0.1653 -8-

MV IS 0.4249 -1- 0.5860 -1- 0.3378 -1- 0.4357 -1- 0.3354 -1- 0.5592 -1-

MV 0.2456 −0.0039 0.1148 0.0071 −0.0554 −0.0580(0.37) -12- (0.00) -17- (0.00) -17- (0.00) -13- (0.01) -16- (0.00) -16-(0.80) (0.00) (0.01) (0.01) (0.32) (0.03)

MinV 0.3725 0.3358 0.3165 0.0804 0.2964 0.2694(0.00) -2- (0.05) -2- (0.14) -3- (0.00) -12- (0.00) -2- (0.04) -3-(0.00) (0.12) (0.41) (0.02) (0.01) (0.14)

VW 0.1593 0.1593 0.1593 0.1593 0.1147 0.1147(0.00) -16- (0.00) -12- (0.00) -15- (0.00) -10- (0.00) -11- (0.00) -11-(0.00) (0.00) (0.00) (0.00) (0.00) (0.00)

BS 0.2993 0.0019 0.2130 0.0040 −0.0501 −0.0579(0.19) -6- (0.00) -16- (0.10) -9- (0.00) -14- (0.01) -14- (0.00) -15-(0.49) (0.01) (0.26) (0.01) (0.41) (0.04)

MVsc 0.2683 0.2627 0.2083 0.2412 0.2162 0.2247(0.33) -9- (0.47) -5- (0.02) -11- (0.35) -8- (0.00) -7- (0.01) -5-(0.72) (0.97) (0.10) (0.73) (0.05) (0.02)

MinVsc 0.2900 0.2859 0.3149 0.3038 0.2400 0.2558(0.04) -7- (0.10) -4- (0.06) -4- (0.04) -2- (0.00) -5- (0.00) -4-(0.13) (0.21) (0.15) (0.10) (0.00) (0.00)

BSsc 0.2731 0.2554 0.2223 0.2578 0.2235 0.2222(0.24) -8- (0.36) -7- (0.05) -8- (0.43) -6- (0.00) -6- (0.01) -6-(0.55) (0.79) (0.12) (0.86) (0.03) (0.01)

MVlsc 0.2409 0.2094 0.1374 0.1155 0.2414 0.2959(0.24) -14- (0.12) -11- (0.00) -16- (0.01) -11- (0.02) -4- (0.02) -2-(0.58) (0.34) (0.02) (0.07) (0.08) (0.09)

MP 0.2178 0.2111 0.2103 0.1928 0.0707 0.0701(0.04) -15- (0.02) -10- (0.01) -10- (0.01) -9- (0.01) -13- (0.01) -12-(0.15) (0.13) (0.09) (0.12) (0.14) (0.13)

MV/MinV 0.3020 0.0092 0.1977 −0.0179 −0.0505 −0.0575(0.17) -5- (0.00) -15- (0.06) -12- (0.00) -16- (0.01) -15- (0.00) -14-(0.43) (0.01) (0.27) (0.01) (0.44) (0.05)

EW/MinV 0.3545 0.3169 0.3195 0.2592 0.2616 0.2087(0.00) -3- (0.00) -3- (0.03) -2- (0.19) -5- (0.00) -3- (0.02) -7-(0.00) (0.02) (0.12) (0.38) (0.01) (0.05)

EW/MV 0.3151 0.0383 0.1632 −0.0201 −0.0601 −0.0591(0.03) -4- (0.00) -13- (0.00) -14- (0.00) -17- (0.01) -17- (0.00) -17-(0.17) (0.02) (0.20) (0.02) (0.44) (0.06)

EW/MV/MinV 0.0858 0.0351 0.1672 −0.0021 0.1086 −0.0541(0.00) -17- (0.00) -14- (0.02) -13- (0.00) -15- (0.28) -12- (0.00) -13-(0.04) (0.01) (0.35) (0.01) (0.62) (0.05)

VT 0.2550 0.2422 0.3052 0.2964 0.1663 0.1533(0.06) -11- (0.01) -8- (0.00) -5- (0.00) -3- (0.31) -8- (0.15) -9-(0.21) (0.02) (0.02) (0.01) (0.69) (0.36)

RRT 0.2433 0.2248 0.2903 0.2880 0.1589 0.1382(0.00) -13- (0.00) -9- (0.03) -6- (0.01) -4- (0.37) -10- (0.02) -10-(0.01) (0.00) (0.12) (0.02) (0.78) (0.07)


Table D.2: Certainty Equivalent Returns, Estimation Window M = 60. This table lists the out-of-sample certainty equivalent returns (CEs) of the naive diversification (EW) approach, the in-sample CEs forthe mean-variance (MV IS) approach, and the out-of-sample CEs of the 15 optimized portfolio strategiesfor all six data sets under consideration. The first number in parentheses represents the p-value of theone-sided JK test on the pairwise difference between the out-of-sample CE of EW and strategy i, i =

MV,MinV, . . . ,RRT. The second number in parentheses represents the p-value of the corresponding two-sided LW bootstrap test. CEs that outperform the EW benchmark at a significance level of α = 0.1 withrespect to LW are highlighted in bold. The numbers indicated by - · - denote the relative ranking of thespecific strategy within each data set based on the value of the CE. All results rely on a rolling-windowapproach with window length M = 60, risk-aversion parameter γ = 1, and tuning parameter η = 1 (onlyVT and RRT).


PF25 USAN = 25

IND10 USAN = 10

IND48 USAN = 48

PF6 EUN = 6

PF25 EUN = 25

EW 0.0114 -12- 0.0118 -6- 0.0105 -9- 0.0108 -5- 0.0069 -9- 0.0071 -8-

MV IS 0.0187 -1- 0.0260 -1- 0.0114 -1- 0.0187 -1- 0.0179 -1- 0.0502 -1-

MV 0.0149 −2.5287 0.0060 −0.5676 −0.3742 −332.9523(0.12) -2- (0.00) -16- (0.11) -17- (0.00) -14- (0.00) -16- (0.00) -16-(0.29) (0.41) (0.18) (0.19) (0.46) (0.46)

MinV 0.0134 0.0134 0.0105 0.0031 0.0115 0.0119(0.09) -5- (0.21) -3- (0.49) -8- (0.01) -12- (0.02) -3- (0.05) -3-(0.22) (0.43) (0.99) (0.04) (0.07) (0.15)

VW 0.0060 0.0060 0.0060 0.0060 0.0045 0.0045(0.00) -16- (0.00) -12- (0.00) -16- (0.00) -10- (0.00) -11- (0.00) -11-(0.00) (0.00) (0.00) (0.00) (0.00) (0.00)

BS 0.0143 −0.5961 0.0094 −0.0898 −0.1984 −191.8042(0.09) -3- (0.00) -14- (0.32) -13- (0.00) -13- (0.00) -14- (0.00) -15-(0.24) (0.45) (0.68) (0.37) (0.45) (0.44)

MVsc 0.0123 0.0124 0.0104 0.0114 0.0099 0.0108(0.13) -8- (0.25) -4- (0.49) -10- (0.35) -2- (0.00) -6- (0.00) -4-(0.32) (0.54) (0.98) (0.71) (0.02) (0.01)

MinVsc 0.0112 0.0109 0.0101 0.0097 0.0096 0.0100(0.39) -14- (0.19) -10- (0.37) -12- (0.20) -8- (0.00) -7- (0.00) -6-(0.79) (0.40) (0.78) (0.42) (0.03) (0.04)

BSsc 0.0122 0.0116 0.0101 0.0098 0.0101 0.0105(0.15) -9- (0.44) -7- (0.41) -11- (0.22) -7- (0.00) -5- (0.00) -5-(0.34) (0.90) (0.79) (0.46) (0.01) (0.01)

MVlsc 0.0122 0.0151 0.0085 0.0057 0.0124 0.0231(0.31) -10- (0.18) -2- (0.30) -14- (0.24) -11- (0.00) -2- (0.00) -2-(0.66) (0.37) (0.59) (0.49) (0.03) (0.00)

MP 0.0097 0.0097 0.0085 0.0085 0.0023 0.0023(0.09) -15- (0.05) -11- (0.04) -15- (0.04) -9- (0.01) -12- (0.01) -12-(0.16) (0.17) (0.16) (0.17) (0.13) (0.12)

MV/MinV 0.0133 −1.2765 0.0120 −27.1123 −0.1784 −77.7802(0.16) -6- (0.00) -15- (0.31) -1- (0.00) -17- (0.00) -13- (0.00) -13-(0.36) (0.20) (0.61) (0.23) (0.47) (0.45)

EW/MinV 0.0128 0.0122 0.0105 0.0105 0.0101 0.0084(0.10) -7- (0.34) -5- (0.47) -7- (0.23) -6- (0.02) -4- (0.10) -7-(0.21) (0.67) (0.96) (0.48) (0.08) (0.28)

EW/MV 0.0140 −0.5287 0.0109 −1.0319 −4.7530 −85.9413(0.03) -4- (0.00) -13- (0.45) -5- (0.00) -15- (0.00) -17- (0.00) -14-(0.10) (0.41) (0.82) (0.35) (0.46) (0.41)

EW/MV/MinV−0.0057 −676.3086 0.0111 −19.9794 −0.3472 −643.8187(0.05) -17- (0.00) -17- (0.44) -4- (0.00) -16- (0.00) -15- (0.00) -17-(0.20) (0.45) (0.70) (0.21) (0.20) (0.44)

VT 0.0114 0.0116 0.0111 0.0113 0.0071 0.0064(0.38) -11- (0.34) -8- (0.06) -3- (0.26) -3- (0.33) -8- (0.13) -9-(0.80) (0.73) (0.16) (0.52) (0.70) (0.30)

RRT 0.0112 0.0111 0.0109 0.0112 0.0067 0.0058(0.32) -13- (0.15) -9- (0.10) -6- (0.27) -4- (0.38) -10- (0.02) -10-(0.61) (0.34) (0.26) (0.57) (0.80) (0.06)


Table D.3: Sharpe Ratios, Estimation Window M = 180. This table lists the out-of-sample Sharpe ra-tios (SRs) of the naive diversification (EW) approach, the in-sample SRs for the mean-variance (MV IS)approach, and the out-of-sample SRs of the 15 optimized portfolio strategies for all six data sets underconsideration. The first number in parentheses represents the p-value of the one-sided JK test on the pair-wise difference between the out-of-sample SR of EW and strategy i, i = MV,MinV, . . . ,RRT. The secondnumber in parentheses represents the p-value of the corresponding two-sided LW bootstrap test. SRs thatoutperform the EW benchmark at a significance level of α = 0.1 with respect to LW are highlighted inbold. The numbers indicated by - · - denote the relative ranking of the specific strategy within each data setbased on the value of the SR. All results rely on a rolling-window approach with window length M = 180,risk-aversion parameter γ = 1, and tuning parameter η = 1 (only VT and RRT).


PF25 USAN = 25

IND10 USAN = 10

IND48 USAN = 48

PF6 EUN = 6

PF25 EUN = 25

EW 0.2276 -10- 0.2259 -10- 0.2548 -11- 0.2315 -6- 0.1260 -11- 0.1289 -6-

MV IS 0.4299 -1- 0.6327 -1- 0.3278 -1- 0.4496 -1- 0.2702 -1- 0.6752 -1-

MV 0.3702 0.4374 0.2239 0.1303 0.1295 0.2252(0.00) -6- (0.00) -5- (0.25) -14- (0.06) -16- (0.48) -9- (0.16) -3-(0.03) (0.01) (0.58) (0.21) (0.97) (0.37)

MinV 0.3750 0.4241 0.3098 0.2241 0.1725 0.1006(0.00) -5- (0.00) -6- (0.08) -3- (0.44) -8- (0.15) -2- (0.31) -13-(0.00) (0.00) (0.25) (0.90) (0.38) (0.59)

VW 0.1640 0.1640 0.1640 0.1640 0.0984 0.0984(0.00) -17- (0.00) -16- (0.00) -16- (0.00) -13- (0.01) -16- (0.02) -14-(0.01) (0.00) (0.00) (0.01) (0.07) (0.12)

BS 0.3849 0.4577 0.2807 0.1854 0.1391 0.2227(0.00) -3- (0.00) -2- (0.26) -7- (0.21) -12- (0.40) -4- (0.15) -4-(0.01) (0.00) (0.61) (0.52) (0.85) (0.33)

MVsc 0.2203 0.2126 0.1950 0.1407 0.1260 0.1070(0.34) -12- (0.27) -13- (0.04) -15- (0.01) -14- (0.50) -12- (0.12) -12-(0.72) (0.61) (0.12) (0.05) (1.00) (0.35)

MinVsc 0.2604 0.2708 0.2940 0.2799 0.1276 0.1237(0.04) -9- (0.02) -9- (0.10) -5- (0.07) -2- (0.46) -10- (0.38) -8-(0.09) (0.02) (0.25) (0.16) (0.93) (0.78)

BSsc 0.2229 0.2129 0.2367 0.1874 0.1254 0.0941(0.39) -11- (0.26) -12- (0.27) -13- (0.10) -11- (0.49) -13- (0.03) -15-(0.81) (0.58) (0.58) (0.29) (0.98) (0.12)

MVlsc 0.2030 0.1537 0.0840 0.0298 0.1173 0.0391(0.21) -15- (0.09) -17- (0.00) -17- (0.00) -17- (0.37) -15- (0.05) -17-(0.49) (0.26) (0.02) (0.02) (0.73) (0.29)

MP 0.2190 0.2160 0.2382 0.2183 0.1250 0.1282(0.00) -13- (0.00) -11- (0.00) -12- (0.00) -9- (0.12) -14- (0.11) -7-(0.01) (0.01) (0.01) (0.00) (0.37) (0.32)

MV/MinV 0.3851 0.4539 0.2874 0.1359 0.1374 0.2157(0.00) -2- (0.00) -3- (0.21) -6- (0.04) -15- (0.41) -5- (0.15) -5-(0.01) (0.00) (0.52) (0.12) (0.87) (0.31)

EW/MinV 0.3647 0.3979 0.3103 0.2578 0.1639 0.1194(0.00) -7- (0.00) -8- (0.04) -2- (0.19) -5- (0.15) -3- (0.39) -9-(0.00) (0.00) (0.15) (0.42) (0.40) (0.76)

EW/MV 0.3770 0.4168 0.2583 0.1934 0.1346 0.2288(0.00) -4- (0.00) -7- (0.44) -10- (0.03) -10- (0.42) -6- (0.07) -2-(0.00) (0.00) (0.91) (0.10) (0.88) (0.17)

EW/MV/MinV 0.3456 0.4469 0.2967 0.2242 −0.0897 0.0735(0.03) -8- (0.00) -4- (0.09) -4- (0.36) -7- (0.03) -17- (0.31) -16-(0.11) (0.00) (0.32) (0.72) (0.11) (0.77)

VT 0.2164 0.1974 0.2792 0.2736 0.1329 0.1142(0.02) -14- (0.00) -14- (0.02) -8- (0.01) -3- (0.19) -7- (0.08) -10-(0.17) (0.03) (0.08) (0.05) (0.47) (0.13)

RRT 0.2019 0.1766 0.2684 0.2732 0.1325 0.1108(0.00) -16- (0.00) -15- (0.08) -9- (0.02) -4- (0.20) -8- (0.04) -11-(0.02) (0.01) (0.23) (0.06) (0.47) (0.08)


Table D.4: Certainty Equivalent Returns, Estimation Window M = 180. This table lists the out-of-sample certainty equivalent returns (CEs) of the naive diversification (EW) approach, the in-sample CEs forthe mean-variance (MV IS) approach, and the out-of-sample CEs of the 15 optimized portfolio strategiesfor all six data sets under consideration. The first number in parentheses represents the p-value of theone-sided JK test on the pairwise difference between the out-of-sample CE of EW and strategy i, i =

MV,MinV, . . . ,RRT. The second number in parentheses represents the p-value of the corresponding two-sided LW bootstrap test. CEs that outperform the EW benchmark at a significance level of α = 0.1 withrespect to LW are highlighted in bold. The numbers indicated by - · - denote the relative ranking of thespecific strategy within each data set based on the value of the CE. All results rely on a rolling-windowapproach with window length M = 180, risk-aversion parameter γ = 1, and tuning parameter η = 1 (onlyVT and RRT).


PF25 USAN = 25

IND10 USAN = 10

IND48 USAN = 48

PF6 EUN = 6

PF25 EUN = 25

EW 0.0096 -13- 0.0098 -11- 0.0096 -9- 0.0095 -5- 0.0055 -13- 0.0057 -6-

MV IS 0.0189 -1- 0.0287 -1- 0.0112 -1- 0.0195 -1- 0.0136 -1- 0.1526 -1-

MV 0.0164 0.0219 0.0084 0.0071 0.0064 0.0190(0.00) -3- (0.00) -2- (0.26) -14- (0.28) -13- (0.41) -5- (0.08) -2-(0.02) (0.00) (0.58) (0.60) (0.86) (0.17)

MinV 0.0138 0.0148 0.0102 0.0073 0.0072 0.0037(0.00) -7- (0.00) -7- (0.34) -3- (0.14) -12- (0.24) -2- (0.25) -15-(0.03) (0.02) (0.71) (0.32) (0.56) (0.47)

VW 0.0062 0.0062 0.0062 0.0062 0.0039 0.0039(0.00) -17- (0.00) -17- (0.00) -16- (0.00) -15- (0.01) -16- (0.02) -13-(0.00) (0.00) (0.00) (0.00) (0.03) (0.10)

BS 0.0155 0.0195 0.0095 0.0078 0.0064 0.0154(0.00) -4- (0.00) -3- (0.47) -10- (0.26) -11- (0.38) -4- (0.08) -3-(0.02) (0.00) (0.96) (0.57) (0.80) (0.21)

MVsc 0.0102 0.0095 0.0079 0.0070 0.0058 0.0048(0.26) -11- (0.37) -13- (0.13) -15- (0.14) -14- (0.38) -10- (0.23) -10-(0.54) (0.73) (0.25) (0.33) (0.79) (0.54)

MinVsc 0.0100 0.0103 0.0097 0.0092 0.0054 0.0050(0.30) -12- (0.31) -9- (0.48) -8- (0.40) -7- (0.44) -15- (0.25) -8-(0.59) (0.58) (0.97) (0.82) (0.89) (0.49)

BSsc 0.0103 0.0094 0.0089 0.0085 0.0058 0.0039(0.23) -10- (0.35) -14- (0.27) -13- (0.26) -8- (0.40) -11- (0.06) -14-(0.51) (0.70) (0.56) (0.57) (0.82) (0.20)

MVlsc 0.0108 0.0099 0.0035 −0.0135 0.0055 −0.0015(0.26) -9- (0.49) -10- (0.04) -17- (0.01) -17- (0.49) -12- (0.11) -16-(0.55) (0.99) (0.10) (0.04) (0.97) (0.34)

MP 0.0094 0.0096 0.0095 0.0094 0.0055 0.0057(0.14) -14- (0.06) -12- (0.29) -11- (0.20) -6- (0.24) -14- (0.24) -7-(0.23) (0.11) (0.55) (0.42) (0.55) (0.56)

MV/MinV 0.0154 0.0178 0.0103 0.0057 0.0063 0.0131(0.00) -5- (0.00) -5- (0.33) -2- (0.08) -16- (0.40) -6- (0.09) -5-(0.02) (0.00) (0.70) (0.17) (0.82) (0.23)

EW/MinV 0.0133 0.0137 0.0101 0.0083 0.0068 0.0046(0.00) -8- (0.00) -8- (0.34) -6- (0.18) -9- (0.25) -3- (0.29) -11-(0.02) (0.02) (0.74) (0.39) (0.56) (0.57)

EW/MV 0.0153 0.0172 0.0090 0.0096 0.0063 0.0135(0.00) -6- (0.00) -6- (0.25) -12- (0.49) -4- (0.39) -7- (0.04) -4-(0.02) (0.00) (0.56) (0.99) (0.84) (0.12)

EW/MV/MinV 0.0170 0.0182 0.0101 0.0083 −5.5939 −50.6224(0.01) -2- (0.00) -4- (0.32) -4- (0.08) -10- (0.00) -17- (0.00) -17-(0.03) (0.00) (0.70) (0.17) (0.29) (0.43)

VT 0.0094 0.0091 0.0101 0.0103 0.0059 0.0048(0.28) -15- (0.10) -15- (0.15) -5- (0.20) -3- (0.21) -8- (0.06) -9-(0.63) (0.24) (0.33) (0.38) (0.43) (0.09)

RRT 0.0090 0.0084 0.0100 0.0105 0.0059 0.0046(0.11) -16- (0.05) -16- (0.16) -7- (0.15) -2- (0.20) -9- (0.03) -12-(0.20) (0.12) (0.41) (0.34) (0.39) (0.05)


Table D.5: Sharpe Ratios, Expanding Window. This table lists the out-of-sample Sharpe ratios (SRs)of the naive diversification (EW) approach, the in-sample SRs for the mean-variance (MV IS) approach,and the out-of-sample SRs of the 15 optimized portfolio strategies for all six data sets under consideration.The first number in parentheses represents the p-value of the one-sided JK test on the pairwise differencebetween the out-of-sample SR of EW and strategy i, i = MV,MinV, . . . ,RRT. The second number inparentheses represents the p-value of the corresponding two-sided LW bootstrap test. SRs that outperformthe EW benchmark at a significance level of α = 0.1 with respect to LW are highlighted in bold. Thenumbers indicated by - · - denote the relative ranking of the specific strategy within each data set basedon the value of the SR. All results rely on an expanding-window approach with initial length M = 120,risk-aversion parameter γ = 1, and tuning parameter η = 1 (only VT and RRT).


PF25 USAN = 25

IND10 USAN = 10

IND48 USAN = 48

PF6 EUN = 6

PF25 EUN = 25

EW 0.2396 -13- 0.2386 -10- 0.2663 -9- 0.2379 -8- 0.1379 -14- 0.1461 -12-

MV IS 0.4460 -1- 0.6351 -1- 0.3399 -1- 0.4350 -1- 0.3896 -1- 0.6324 -1-

MV 0.3777 0.3911 0.2456 0.1949 0.2013 0.2100(0.00) -4- (0.00) -3- (0.32) -13- (0.23) -11- (0.13) -6- (0.22) -3-(0.04) (0.04) (0.70) (0.55) (0.39) (0.51)

MinV 0.3601 0.3843 0.3096 0.2416 0.2390 0.1702(0.00) -6- (0.00) -4- (0.12) -3- (0.47) -7- (0.00) -3- (0.29) -6-(0.04) (0.04) (0.70) (0.55) (0.39) (0.51)

VW 0.1525 0.1525 0.1525 0.1525 0.0964 0.0964(0.00) -17- (0.00) -17- (0.00) -16- (0.00) -14- (0.00) -16- (0.00) -17-(0.00) (0.00) (0.00) (0.00) (0.01) (0.01)

BS 0.3871 0.4322 0.3000 0.2469 0.2342 0.2266(0.00) -2- (0.00) -2- (0.18) -4- (0.43) -6- (0.02) -4- (0.12) -2-(0.02) (0.00) (0.43) (0.86) (0.08) (0.30)

MVsc 0.2624 0.2404 0.1948 0.1861 0.1653 0.1373(0.08) -10- (0.46) -9- (0.01) -15- (0.09) -12- (0.05) -10- (0.31) -14-(0.18) (0.93) (0.07) (0.27) (0.26) (0.67)

MinVsc 0.2436 0.2544 0.2959 0.3007 0.1738 0.1767(0.42) -12- (0.25) -6- (0.13) -5- (0.02) -2- (0.00) -8- (0.01) -5-(0.83) (0.51) (0.30) (0.08) (0.03) (0.07)

BSsc 0.2579 0.2481 0.2512 0.2295 0.1666 0.1483(0.12) -11- (0.27) -7- (0.29) -12- (0.40) -9- (0.04) -9- (0.45) -11-(0.27) (0.59) (0.57) (0.82) (0.24) (0.92)

MVlsc 0.2648 0.2224 0.1359 0.0667 0.1584 0.1604(0.21) -8- (0.38) -13- (0.01) -17- (0.00) -15- (0.21) -11- (0.40) -9-(0.48) (0.79) (0.05) (0.03) (0.49) (0.87)

MP 0.2320 0.2300 0.2563 0.2284 0.1350 0.1432(0.00) -14- (0.00) -12- (0.00) -10- (0.00) -10- (0.00) -15- (0.00) -13-(0.01) (0.03) (0.00) (0.00) (0.01) (0.03)

MV/MinV 0.3780 0.2026 0.2135 −0.0387 0.2467 0.1625(0.00) -3- (0.21) -15- (0.11) -14- (0.00) -17- (0.01) -2- (0.38) -8-(0.01) (0.46) (0.26) (0.00) (0.04) (0.74)

EW/MinV 0.3499 0.3547 0.3147 0.2712 0.1997 0.1537(0.00) -7- (0.00) -5- (0.02) -2- (0.01) -5- (0.00) -7- (0.23) -10-(0.00) (0.00) (0.09) (0.03) (0.02) (0.49)

EW/MV 0.3761 0.2371 0.2529 0.0615 0.2016 0.1683(0.00) -5- (0.46) -11- (0.06) -11- (0.00) -16- (0.03) -5- (0.18) -7-(0.00) (0.93) (0.22) (0.00) (0.12) (0.50)

EW/MV/MinV 0.2647 0.2410 0.2680 0.1761 0.0770 0.2055(0.28) -9- (0.46) -8- (0.47) -8- (0.00) -13- (0.24) -17- (0.12) -4-(0.76) (0.92) (0.95) (0.01) (0.63) (0.24)

VT 0.2246 0.2059 0.2865 0.2879 0.1401 0.1268(0.00) -15- (0.00) -14- (0.01) -6- (0.00) -3- (0.38) -12- (0.02) -15-(0.02) (0.01) (0.07) (0.01) (0.79) (0.03)

RRT 0.2223 0.1981 0.2824 0.2858 0.1395 0.1181(0.00) -16- (0.00) -16- (0.03) -7- (0.00) -4- (0.41) -13- (0.00) -16-(0.03) (0.01) (0.12) (0.01) (0.82) (0.01)


Table D.6: Certainty Equivalent Returns, Expanding Window. This table lists the out-of-sample cer-tainty equivalent returns (CEs) of the naive diversification (EW) approach, the in-sample CEs for the mean-variance (MV IS) approach, and the out-of-sample CEs of the 15 optimized portfolio strategies for all sixdata sets under consideration. The first number in parentheses represents the p-value of the one-sided JK teston the pairwise difference between the out-of-sample CE of EW and strategy i, i = MV,MinV, . . . ,RRT.The second number in parentheses represents the p-value of the corresponding two-sided LW bootstraptest. CEs that outperform the EW benchmark at a significance level of α = 0.1 with respect to LW arehighlighted in bold. The numbers indicated by - · - denote the relative ranking of the specific strategy withineach data set based on the value of the CE. All results rely on an expanding-window approach with initiallength M = 120, risk-aversion parameter γ = 1, and tuning parameter η = 1 (only VT and RRT).


PF25 USAN = 25

IND10 USAN = 10

IND48 USAN = 48

PF6 EUN = 6

PF25 EUN = 25

EW 0.0102 -12- 0.0104 -10- 0.0101 -9- 0.0099 -6- 0.0060 -14- 0.0065 -12-

MV IS 0.0196 -1- 0.0281 -1- 0.0114 -1- 0.0173 -1- 0.0285 -1- 0.0900 -1-

MV 0.0175 0.0198 0.0099 0.0112 0.0111 0.0164(0.00) -2- (0.00) -2- (0.46) -13- (0.36) -2- (0.07) -4- (0.08) -2-(0.01) (0.00) (0.95) (0.75) (0.21) (0.21)

MinV 0.0136 0.0136 0.0105 0.0083 0.0100 0.0068(0.01) -7- (0.04) -5- (0.37) -6- (0.21) -12- (0.01) -5- (0.45) -9-(0.05) (0.09) (0.77) (0.46) (0.05) (0.91)

VW 0.0057 0.0057 0.0057 0.0057 0.0037 0.0037(0.00) -17- (0.00) -17- (0.00) -17- (0.00) -14- (0.00) -16- (0.00) -17-(0.00) (0.00) (0.00) (0.00) (0.00) (0.00)

BS 0.0155 0.0163 0.0105 0.0093 0.0113 0.0132(0.00) -3- (0.00) -4- (0.39) -8- (0.39) -10- (0.02) -3- (0.06) -3-(0.02) (0.00) (0.82) (0.81) (0.09) (0.20)

MVsc 0.0123 0.0115 0.0082 0.0090 0.0078 0.0063(0.00) -9- (0.13) -7- (0.10) -15- (0.33) -11- (0.03) -10- (0.42) -14-(0.02) (0.33) (0.23) (0.67) (0.19) (0.85)

MinVsc 0.0095 0.0099 0.0099 0.0101 0.0075 0.0075(0.23) -16- (0.31) -12- (0.45) -12- (0.45) -5- (0.01) -11- (0.11) -7-(0.45) (0.59) (0.89) (0.91) (0.05) (0.24)

BSsc 0.0119 0.0113 0.0096 0.0093 0.0079 0.0067(0.01) -11- (0.13) -8- (0.35) -14- (0.34) -9- (0.02) -9- (0.41) -10-(0.05) (0.30) (0.69) (0.71) (0.18) (0.84)

MVlsc 0.0138 0.0180 0.0066 −0.0028 0.0079 0.0112(0.02) -6- (0.05) -3- (0.13) -16- (0.06) -16- (0.11) -8- (0.20) -4-(0.04) (0.19) (0.29) (0.13) (0.34) (0.52)

MP 0.0101 0.0103 0.0100 0.0098 0.0059 0.0064(0.24) -13- (0.13) -11- (0.40) -10- (0.12) -8- (0.00) -15- (0.00) -13-(0.55) (0.32) (0.80) (0.30) (0.01) (0.03)

MV/MinV 0.0145 0.0085 0.0107 −0.0242 0.0115 0.0070(-0.01) -5- (0.18) -16- (0.39) -2- (0.00) -17- (0.01) -2- (0.43) -8-(0.03) (0.37) (0.78) (0.00) (0.05) (0.85)

EW/MinV 0.0130 0.0126 0.0105 0.0099 0.0084 0.0065(0.01) -8- (0.03) -6- (0.33) -7- (0.48) -7- (0.02) -7- (0.50) -11-(0.05) (0.08) (0.71) (0.97) (0.06) (0.99)

EW/MV 0.0149 0.0110 0.0100 −0.0001 0.0096 0.0080(0.00) -4- (0.21) -9- (0.44) -11- (0.02) -15- (0.03) -6- (0.14) -6-(0.00) (0.39) (0.88) (0.03) (0.09) (0.41)

EW/MV/MinV 0.0122 0.0092 0.0107 0.0073 −0.3003 0.0097(0.17) -10- (0.13) -15- (0.29) -3- (0.00) -13- (0.00) -17- (0.12) -5-(0.43) (0.26) (0.59) (0.01) (0.42) (0.29)

VT 0.0099 0.0096 0.0106 0.0109 0.0061 0.0054(0.11) -15- (0.05) -13- (0.08) -4- (0.10) -4- (0.37) -12- (0.01) -15-(0.25) (0.14) (0.21) (0.20) (0.74) (0.02)

RRT 0.0099 0.0094 0.0105 0.0110 0.0061 0.0049(0.25) -14- (0.05) -14- (0.08) -5- (0.08) -3- (0.39) -13- (0.00) -16-(0.50) (0.11) (0.21) (0.16) (0.80) (0.00)


Table D.7: Sharpe Ratios, Risk-Aversion Parameter γ = 2. This table lists the out-of-sample Sharperatios (SRs) of the naive diversification (EW) approach, the in-sample SRs for the mean-variance (MV IS)approach, and the out-of-sample SRs of the 15 optimized portfolio strategies for all six data sets underconsideration. The first number in parentheses represents the p-value of the one-sided JK test on the pair-wise difference between the out-of-sample SR of EW and strategy i, i = MV,MinV, . . . ,RRT. The secondnumber in parentheses represents the p-value of the corresponding two-sided LW bootstrap test. SRs thatoutperform the EW benchmark at a significance level of α = 0.1 with respect to LW are highlighted inbold. The numbers indicated by - · - denote the relative ranking of the specific strategy within each data setbased on the value of the SR. All results rely on a rolling-window approach with window length M = 120,risk-aversion parameter γ = 2, and tuning parameter η = 1 (only VT and RRT).


PF25 USAN = 25

IND10 USAN = 10

IND48 USAN = 48

PF6 EUN = 6

PF25 EUN = 25

EW 0.2396 -12- 0.2386 -12- 0.2663 -9- 0.2379 -6- 0.1379 -14- 0.1461 -11-

MV IS 0.4460 -1- 0.6351 -1- 0.3399 -1- 0.4350 -1- 0.3896 -1- 0.6324 -1-

MV 0.3912 0.3942 0.2208 0.1037 0.1861 0.2227(0.00) -6- (0.00) -7- (0.17) -13- (0.02) -15- (0.23) -8- (0.20) -4-(0.01) (0.04) (0.46) (0.09) (0.58) (0.48)

MinV 0.3964 0.4302 0.3194 0.2078 0.2219 0.1383(0.00) -4- (0.00) -5- (0.08) -3- (0.27) -8- (0.02) -2- (0.44) -14-(0.00) (0.00) (0.23) (0.53) (0.10) (0.87)

VW 0.1525 0.1525 0.1525 0.1525 0.0964 0.0964(0.00) -16- (0.00) -17- (0.00) -16- (0.00) -11- (0.00) -17- (0.00) -17-(0.00) (0.00) (0.00) (0.00) (0.00) (0.01)

BS 0.4122 0.4682 0.2827 0.1564 0.2040 0.2261(0.00) -2- (0.00) -3- (0.34) -8- (0.08) -10- (0.13) -3- (0.18) -3-(0.00) (0.00) (0.73) (0.24) (0.36) (0.43)

MVsc 0.2640 0.2658 0.1653 0.1342 0.1688 0.1627(0.08) -11- (0.08) -10- (0.00) -15- (0.00) -12- (0.06) -11- (0.18) -9-(0.20) (0.24) (0.02) (0.03) (0.26) (0.41)

MinVsc 0.2665 0.2756 0.3091 0.2823 0.1802 0.1803(0.07) -10- (0.03) -9- (0.07) -4- (0.08) -4- (0.01) -9- (0.05) -8-(0.15) (0.04) (0.18) (0.18) (0.10) (0.22)

BSsc 0.2683 0.2605 0.2010 0.1780 0.1700 0.1595(0.05) -9- (0.11) -11- (0.02) -14- (0.04) -9- (0.04) -10- (0.23) -10-(0.14) (0.32) (0.06) (0.13) (0.26) (0.52)

MVlsc 0.2793 0.2278 0.0953 0.0556 0.1874 0.2033(0.09) -8- (0.42) -13- (0.00) -17- (0.00) -16- (0.06) -7- (0.16) -7-(0.21) (0.85) (0.00) (0.02) (0.30) (0.46)

MP 0.2302 0.2270 0.2351 0.2233 0.1343 0.1420(0.00) -13- (0.00) -14- (0.00) -12- (0.00) -7- (0.00) -15- (0.00) -12-(0.03) (0.03) (0.04) (0.00) (0.02) (0.01)

MV/MinV 0.4122 0.4712 0.2630 0.0476 0.1946 0.2201(0.00) -3- (0.00) -2- (0.47) -10- (0.00) -17- (0.15) -5- (0.18) -5-(0.00) (0.00) (0.95) (0.01) (0.43) (0.44)

EW/MinV 0.3826 0.3938 0.3215 0.2648 0.1988 0.1404(0.00) -7- (0.00) -8- (0.03) -2- (0.13) -5- (0.03) -4- (0.42) -13-(0.00) (0.00) (0.09) (0.27) (0.13) (0.83)

EW/MV 0.3915 0.3977 0.2567 0.1232 0.1932 0.2321(0.00) -5- (0.00) -6- (0.27) -11- (0.00) -13- (0.11) -6- (0.10) -2-(0.00) (0.00) (0.57) (0.00) (0.35) (0.24)

EW/MV/MinV 0.0506 0.4475 0.2964 0.1061 0.0982 0.2120(0.00) -17- (0.00) -4- (0.15) -6- (0.00) -14- (0.29) -16- (0.23) -6-(0.03) (0.00) (0.37) (0.05) (0.61) (0.52)

VT 0.2282 0.2102 0.2972 0.2886 0.1422 0.1287(0.01) -14- (0.00) -15- (0.00) -5- (0.00) -2- (0.27) -12- (0.03) -15-(0.10) (0.00) (0.03) (0.01) (0.56) (0.05)

RRT 0.2178 0.1934 0.2862 0.2879 0.1412 0.1211(0.00) -15- (0.00) -16- (0.02) -7- (0.00) -3- (0.32) -13- (0.01) -16-(0.01) (0.01) (0.09) (0.01) (0.65) (0.01)


Table D.8: Certainty Equivalent Returns, Risk-Aversion Parameter γ = 2. This table lists the out-of-sample certainty equivalent returns (CEs) of the naive diversification (EW) approach, the in-sampleCEs for the mean-variance (MV IS) approach, and the out-of-sample CEs of the 15 optimized portfoliostrategies for all six data sets under consideration. The first number in parentheses represents the p-valueof the one-sided JK test on the pairwise difference between the out-of-sample CE of EW and strategy i,i = MV,MinV, . . . ,RRT. The second number in parentheses represents the p-value of the correspondingtwo-sided LW bootstrap test. CEs that outperform the EW benchmark at a significance level of α = 0.1with respect to LW are highlighted in bold. The numbers indicated by - · - denote the relative ranking of thespecific strategy within each data set based on the value of the CE. All results rely on an expanding-windowapproach with initial length M = 120, risk-aversion parameter γ = 2, and tuning parameter η = 1 (only VTand RRT).


PF25 USAN = 25

IND10 USAN = 10

IND48 USAN = 48

PF6 EUN = 6

PF25 EUN = 25

EW 0.0090 -12- 0.0092 -13- 0.0092 -8- 0.0088 -4- 0.0045 -14- 0.0050 -10-

MV IS 0.0185 -1- 0.0271 -1- 0.0107 -1- 0.0165 -1- 0.0253 -1- 0.0788 -1-

MV 0.0171 0.0209 0.0082 −0.0010 0.0081 0.0019(0.00) -2- (0.00) -2- (0.31) -13- (0.04) -15- (0.20) -4- (0.43) -17-(0.00) (0.00) (0.67) (0.12) (0.49) (0.81)

MinV 0.0137 0.0147 0.0099 0.0067 0.0083 0.0044(0.00) -6- (0.00) -7- (0.33) -3- (0.16) -9- (0.04) -3- (0.41) -13-(0.00) (0.01) (0.69) (0.34) (0.15) (0.82)

VW 0.0048 0.0048 0.0048 0.0048 0.0023 0.0023(0.00) -16- (0.00) -17- (0.00) -16- (0.00) -11- (0.00) -16- (0.00) -16-(0.00) (0.00) (0.00) (0.00) (0.00) (0.00)

BS 0.0160 0.0199 0.0092 0.0059 0.0085 0.0113(0.00) -3- (0.00) -3- (0.49) -9- (0.18) -10- (0.11) -2- (0.29) -4-(0.00) (0.00) (0.99) (0.43) (0.33) (0.51)

MVsc 0.0107 0.0107 0.0063 0.0045 0.0063 0.0060(0.03) -10- (0.06) -10- (0.07) -15- (0.04) -12- (0.05) -11- (0.16) -8-(0.08) (0.20) (0.16) (0.14) (0.24) (0.37)

MinVsc 0.0095 0.0097 0.0095 0.0087 0.0066 0.0065(0.31) -11- (0.32) -12- (0.42) -6- (0.45) -5- (0.02) -9- (0.10) -7-(0.63) (0.65) (0.84) (0.88) (0.10) (0.30)

BSsc 0.0109 0.0104 0.0077 0.0069 0.0064 0.0058(0.02) -9- (0.10) -11- (0.16) -14- (0.15) -8- (0.04) -10- (0.21) -9-(0.06) (0.26) (0.29) (0.34) (0.21) (0.47)

MVlsc 0.0129 0.0125 0.0005 −0.0337 0.0077 0.0103(0.01) -8- (0.21) -9- (0.01) -17- (0.00) -17- (0.05) -7- (0.16) -5-(0.03) (0.45) (0.02) (0.00) (0.27) (0.46)

MP 0.0088 0.0089 0.0084 0.0086 0.0043 0.0048(0.09) -13- (0.04) -14- (0.05) -12- (0.09) -6- (0.00) -15- (0.00) -11-(0.16) (0.08) (0.16) (0.22) (0.02) (0.01)

MV/MinV 0.0156 0.0188 0.0087 −0.0080 0.0078 0.0121(0.00) -4- (0.00) -4- (0.39) -11- (0.00) -16- (0.15) -5- (0.18) -3-(0.00) (0.00) (0.81) (0.00) (0.39) (0.34)

EW/MinV 0.0131 0.0132 0.0098 0.0085 0.0072 0.0045(0.00) -7- (0.00) -8- (0.30) -5- (0.37) -7- (0.06) -8- (0.37) -12-(0.00) (0.00) (0.64) (0.72) (0.19) (0.73)

EW/MV 0.0152 0.0170 0.0089 0.0026 0.0078 0.0129(0.00) -5- (0.00) -5- (0.29) -10- (0.04) -13- (0.10) -6- (0.08) -2-(0.00) (0.00) (0.62) (0.02) (0.28) (0.17)

EW/MV/MinV −7.5672 0.0168 0.0093 0.0015 −0.0026 0.0092(0.00) -17- (0.00) -6- (0.49) -7- (0.03) -14- (0.17) -17- (0.36) -6-(0.47) (0.00) (0.97) (0.16) (0.33) (0.70)

VT 0.0088 0.0083 0.0100 0.0100 0.0048 0.0040(0.15) -14- (0.03) -15- (0.05) -2- (0.08) -3- (0.27) -12- (0.03) -14-(0.34) (0.09) (0.15) (0.17) (0.59) (0.05)

RRT 0.0085 0.0077 0.0098 0.0101 0.0047 0.0036(0.08) -15- (0.02) -16- (0.06) -4- (0.06) -2- (0.32) -13- (0.01) -15-(0.15) (0.03) (0.16) (0.12) (0.63) (0.01)


Table D.9: Sharpe Ratios, Risk-Aversion Parameter γ = 4. This table lists the out-of-sample Sharperatios (SRs) of the naive diversification (EW) approach, the in-sample SRs for the mean-variance (MV IS)approach, and the out-of-sample SRs of the 15 optimized portfolio strategies for all six data sets underconsideration. The first number in parentheses represents the p-value of the one-sided JK test on the pair-wise difference between the out-of-sample SR of EW and strategy i, i = MV,MinV, . . . ,RRT. The secondnumber in parentheses represents the p-value of the corresponding two-sided LW bootstrap test. SRs thatoutperform the EW benchmark at a significance level of α = 0.1 with respect to LW are highlighted inbold. The numbers indicated by - · - denote the relative ranking of the specific strategy within each data setbased on the value of the SR. All results rely on a rolling-window approach with window length M = 120,risk-aversion parameter γ = 4, and tuning parameter η = 1 (only VT and RRT).


PF25 USAN = 25

IND10 USAN = 10

IND48 USAN = 48

PF6 EUN = 6

PF25 EUN = 25

EW 0.2396 -12- 0.2386 -12- 0.2663 -9- 0.2379 -6- 0.1379 -14- 0.1461 -11-

MV IS 0.4460 -1- 0.6351 -1- 0.3399 -1- 0.4350 -1- 0.3896 -1- 0.6324 -1-

MV 0.3912 0.3942 0.2208 0.1037 0.1861 0.2227(0.00) -5- (0.00) -6- (0.17) -13- (0.02) -15- (0.23) -8- (0.20) -4-(0.01) (0.04) (0.44) (0.08) (0.57) (0.48)

MinV 0.3964 0.4302 0.3194 0.2078 0.2219 0.1383(0.00) -4- (0.00) -5- (0.08) -3- (0.27) -9- (0.02) -2- (0.44) -14-(0.00) (0.00) (0.23) (0.58) (0.09) (0.89)

VW 0.1525 0.1525 0.1525 0.1525 0.0964 0.0964(0.00) -16- (0.00) -17- (0.00) -16- (0.00) -12- (0.00) -16- (0.00) -17-(0.00) (0.00) (0.00) (0.00) (0.00) (0.01)

BS 0.4122 0.4682 0.2827 0.1564 0.2040 0.2261(0.00) -2- (0.00) -3- (0.34) -8- (0.08) -11- (0.13) -3- (0.18) -3-(0.00) (0.00) (0.75) (0.24) (0.34) (0.41)

MVsc 0.2640 0.2658 0.1653 0.1342 0.1688 0.1627(0.08) -11- (0.08) -10- (0.00) -15- (0.00) -13- (0.06) -11- (0.18) -9-(0.19) (0.26) (0.02) (0.03) (0.26) (0.37)

MinVsc 0.2665 0.2756 0.3091 0.2823 0.1802 0.1803(0.07) -10- (0.03) -9- (0.07) -4- (0.08) -4- (0.01) -9- (0.05) -8-(0.16) (0.05) (0.20) (0.16) (0.09) (0.21)

BSsc 0.2683 0.2605 0.2010 0.1780 0.1700 0.1595(0.05) -9- (0.11) -11- (0.02) -14- (0.04) -10- (0.04) -10- (0.23) -10-(0.14) (0.33) (0.07) (0.16) (0.25) (0.53)

MVlsc 0.2793 0.2278 0.0953 0.0556 0.1874 0.2033(0.09) -8- (0.42) -13- (0.00) -17- (0.00) -16- (0.06) -7- (0.16) -7-(0.21) (0.86) (0.00) (0.03) (0.30) (0.46)

MP 0.2302 0.2270 0.2351 0.2233 0.1343 0.1420(0.00) -13- (0.00) -14- (0.00) -12- (0.00) -7- (0.00) -15- (0.00) -12-(0.03) (0.02) (0.04) (0.00) (0.03) (0.02)

MV/MinV 0.4122 0.4712 0.2630 0.0476 0.1946 0.2201(0.00) -3- (0.00) -2- (0.47) -10- (0.00) -17- (0.15) -5- (0.18) -5-(0.00) (0.00) (0.95) (0.00) (0.42) (0.41)

EW/MinV 0.3826 0.3938 0.3215 0.2648 0.1988 0.1404(0.00) -7- (0.00) -7- (0.03) -2- (0.13) -5- (0.03) -4- (0.42) -13-(0.00) (0.00) (0.07) (0.26) (0.14) (0.84)

EW/MV 0.3861 0.3932 0.2458 0.1248 0.1883 0.2293(0.00) -6- (0.00) -8- (0.13) -11- (0.00) -14- (0.14) -6- (0.12) -2-(0.00) (0.00) (0.29) (0.00) (0.36) (0.30)

EW/MV/MinV 0.0785 0.4591 0.2947 0.2087 −0.0714 0.2191(0.01) -17- (0.00) -4- (0.16) -6- (0.14) -8- (0.01) -17- (0.20) -6-(0.02) (0.00) (0.42) (0.31) (0.12) (0.48)

VT 0.2282 0.2102 0.2972 0.2886 0.1422 0.1287(0.01) -14- (0.00) -15- (0.00) -5- (0.00) -2- (0.27) -12- (0.03) -15-(0.10) (0.00) (0.03) (0.01) (0.60) (0.07)

RRT 0.2178 0.1934 0.2862 0.2879 0.1412 0.1211(0.00) -15- (0.00) -16- (0.02) -7- (0.00) -3- (0.32) -13- (0.01) -16-(0.01) (0.00) (0.09) (0.01) (0.67) (0.01)


Table D.10: Certainty Equivalent Returns, Risk-Aversion Parameter γ = 4. This table lists the out-of-sample certainty equivalent returns (CEs) of the naive diversification (EW) approach, the in-sampleCEs for the mean-variance (MV IS) approach, and the out-of-sample CEs of the 15 optimized portfoliostrategies for all six data sets under consideration. The first number in parentheses represents the p-valueof the one-sided JK test on the pairwise difference between the out-of-sample CE of EW and strategy i,i = MV,MinV, . . . ,RRT. The second number in parentheses represents the p-value of the correspondingtwo-sided LW bootstrap test. CEs that outperform the EW benchmark at a significance level of α = 0.1with respect to LW are highlighted in bold. The numbers indicated by - · - denote the relative ranking of thespecific strategy within each data set based on the value of the CE. All results rely on an expanding-windowapproach with initial length M = 120, risk-aversion parameter γ = 4, and tuning parameter η = 1 (only VTand RRT).


PF25 USAN = 25

IND10 USAN = 10

IND48 USAN = 48

PF6 EUN = 6

PF25 EUN = 25

EW 0.0068 -12- 0.0069 -12- 0.0075 -9- 0.0067 -6- 0.0016 -14- 0.0020 -7-

MV IS 0.0164 -1- 0.0251 -1- 0.0095 -1- 0.0150 -1- 0.0189 -1- 0.0565 -1-

MV 0.0146 0.0169 0.0060 −0.0136 0.0032 −0.0438(0.00) -2- (0.00) -3- (0.23) -13- (0.00) -15- (0.36) -10- (0.01) -17-(0.00) (0.01) (0.51) (0.01) (0.77) (0.02)

MinV 0.0123 0.0133 0.0087 0.0051 0.0060 0.0019(0.00) -6- (0.00) -7- (0.23) -2- (0.23) -9- (0.02) -2- (0.49) -8-(0.00) (0.00) (0.50) (0.46) (0.11) (0.97)

VW 0.0028 0.0028 0.0028 0.0028 −0.0006 −0.0006(0.00) -16- (0.00) -17- (0.00) -15- (0.00) -11- (0.00) -16- (0.00) -14-(0.00) (0.00) (0.00) (0.00) (0.00) (0.01)

BS 0.0141 0.0177 0.0078 0.0019 0.0051 −0.0115(0.00) -3- (0.00) -2- (0.44) -8- (0.07) -12- (0.16) -3- (0.14) -16-(0.00) (0.00) (0.89) (0.22) (0.39) (0.21)

MVsc 0.0082 0.0082 0.0027 −0.0010 0.0032 0.0028(0.06) -10- (0.08) -9- (0.01) -16- (0.00) -13- (0.09) -11- (0.24) -4-(0.15) (0.21) (0.04) (0.02) (0.35) (0.54)

MinVsc 0.0077 0.0080 0.0083 0.0074 0.0040 0.0040(0.16) -11- (0.14) -10- (0.26) -5- (0.31) -4- (0.01) -7- (0.05) -2-(0.34) (0.28) (0.57) (0.62) (0.04) (0.18)

BSsc 0.0084 0.0080 0.0050 0.0036 0.0033 0.0026(0.04) -9- (0.12) -11- (0.06) -14- (0.05) -10- (0.07) -9- (0.29) -5-(0.09) (0.30) (0.10) (0.15) (0.31) (0.63)

MVlsc 0.0095 0.0040 −0.0075 −0.0792 0.0040 0.0011(0.07) -8- (0.26) -16- (0.00) -17- (0.00) -17- (0.12) -8- (0.44) -10-(0.13) (0.52) (0.00) (0.00) (0.44) (0.91)

MP 0.0065 0.0064 0.0065 0.0061 0.0014 0.0018(0.01) -13- (0.00) -13- (0.01) -12- (0.01) -7- (0.00) -15- (0.00) -9-(0.04) (0.02) (0.08) (0.05) (0.01) (0.00)

MV/MinV 0.0138 0.0169 0.0072 −0.0216 0.0046 0.0000(0.00) -4- (0.00) -4- (0.42) -10- (0.00) -16- (0.18) -5- (0.41) -13-(0.00) (0.00) (0.87) (0.00) (0.44) (0.79)

EW/MinV 0.0116 0.0119 0.0086 0.0071 0.0049 0.0021(0.00) -7- (0.00) -8- (0.17) -3- (0.37) -5- (0.03) -4- (0.49) -6-(0.00) (0.00) (0.41) (0.76) (0.10) (0.97)

EW/MV 0.0131 0.0143 0.0069 −0.0046 0.0042 0.0034(0.00) -5- (0.00) -6- (0.19) -11- (0.00) -14- (0.17) -6- (0.42) -3-(0.00) (0.00) (0.39) (0.00) (0.45) (0.80)

EW/MV/MinV −0.2084 0.0154 0.0079 0.0054 −1.1525 −0.0051(0.00) -17- (0.00) -5- (0.37) -7- (0.17) -8- (0.00) -17- (0.25) -15-(0.34) (0.00) (0.76) (0.33) (0.16) (0.43)

VT 0.0064 0.0055 0.0085 0.0084 0.0018 0.0011(0.04) -14- (0.00) -14- (0.03) -4- (0.03) -3- (0.28) -12- (0.05) -11-(0.13) (0.01) (0.07) (0.05) (0.61) (0.09)

RRT 0.0059 0.0046 0.0082 0.0084 0.0018 0.0007(0.01) -15- (0.00) -15- (0.04) -6- (0.02) -2- (0.33) -13- (0.01) -12-(0.02) (0.00) (0.07) (0.14) (0.71) (0.02)


Table D.11: Sharpe Ratios and Certainty Equivalent Returns, Tuning Parameter η = 2 and η = 4.This table lists the out-of-sample Sharpe ratios (SRs) and certainty equivalent returns (CEs) of the naivediversification (EW) approach and the out-of-sample SRs and CEs of the VT and RRT strategy for tuningparameter values of η = 1, η = 2, and η = 4 for all six data sets under consideration, respectively. The firstnumber in parentheses represents the p-value of the one-sided JK test on the pairwise difference betweenthe out-of-sample SR or CE of EW and strategy i, i = VT,RRT. The second number in parenthesesrepresents the p-value of the corresponding two-sided LW bootstrap test. SRs and CEs that outperform theEW benchmark at a significance level of α = 0.1 with respect to LW are highlighted in bold. All resultsrely on a rolling-window approach with window length M = 120 and risk-aversion parameter γ = 1.

Sharpe Ratios


PF25 USAN = 25

IND10 USAN = 10

IND48 USAN = 48

PF6 EUN = 6

PF25 EUN = 25

EW 0.2396 0.2386 0.2663 0.2379 0.1379 0.1461VTη=1 0.2282 0.2102 0.2972 0.2886 0.1422 0.1287

(0.01) (0.00) (0.00) (0.00) (0.27) (0.03)(0.10) (0.00) (0.03) (0.01) (0.56) (0.06)

RTTη=1 0.2178 0.1934 0.2862 0.2879 0.1412 0.1211(0.00) (0.00) (0.02) (0.00) (0.32) (0.01)(0.02) (0.00) (0.10) (0.01) (0.68) (0.02)

VTη=2 0.2459 0.2353 0.3133 0.3030 0.1525 0.1431(0.04) (0.30) (0.00) (0.00) (0.02) (0.37)(0.21) (0.66) (0.02) (0.00) (0.12) (0.75)

RTTη=2 0.2308 0.2092 0.2976 0.3021 0.1520 0.1310(0.07) (0.01) (0.00) (0.00) (0.02) (0.05)(0.23) (0.04) (0.04) (0.00) (0.09) (0.11)

VTη=4 0.2626 0.2627 0.3257 0.3201 0.1683 0.1628(0.00) (0.00) (0.01) (0.00) (0.00) (0.05)(0.01) (0.02) (0.02) (0.00) (0.05) (0.22)

RRTη=4 0.2516 0.2346 0.3125 0.3202 0.1733 0.1538(0.01) (0.37) (0.00) (0.00) (0.00) (0.21)(0.08) (0.82) (0.03) (0.00) (0.02) (0.49)

Certainty Equivalent Returns


PF25 USAN = 25

IND10 USAN = 10

IND48 USAN = 48

PF6 EUN = 6

PF25 EUN = 25

EW 0.0102 0.0104 0.0101 0.0099 0.0060 0.0065VTη=1 0.0100 0.0097 0.0107 0.0108 0.0062 0.0055

(0.25) (0.08) (0.07) (0.14) (0.27) (0.02)(0.55) (0.20) (0.19) (0.29) (0.57) (0.04)

RRTη=1 0.0098 0.0092 0.0106 0.0110 0.0062 0.0051(0.18) (0.05) (0.07) (0.09) (0.31) (0.00)(0.35) (0.09) (0.16) (0.18) (0.66) (0.01)

VTη=2 0.0103 0.0104 0.0109 0.0109 0.0067 0.0062(0.16) (0.48) (0.09) (0.15) (0.03) (0.30)(0.42) (0.97) (0.19) (0.31) (0.09) (0.57)

RRTη=2 0.0102 0.0098 0.0108 0.0113 0.0067 0.0056(0.50) (0.19) (0.06) (0.07) (0.03) (0.04)(0.99) (0.39) (0.17) (0.16) (0.05) (0.07)

VTη=4 0.0105 0.0109 0.0109 0.0110 0.0075 0.0072(0.23) (0.08) (0.17) (0.19) (0.00) (0.09)(0.46) (0.26) (0.36) (0.37) (0.02) (0.21)

RRTη=4 0.0107 0.0107 0.0110 0.0117 0.0078 0.0068(0.02) (0.31) (0.07) (0.06) (0.00) (0.29)(0.07) (0.68) (0.21) (0.11) (0.00) (0.57)

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